A Friedman number is a positive integer which can be written with the help of symbols like - + , - , x, / , (), ^ etc. and using the digits of the numbers.
Example:-
25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024,
1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349,
2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592 ,2737, 2916, 3125,
3159, 3281, 3375, 3378, 3685, 3784, 3864, 3972, 4088, 4096, 4106, 4167, 4536, 4624,
4628, 5120, 5776, 5832, 6144, 6145, 6455, 6880, 7928, 8092, 8192, 9025, 9216, 9261.
Explanation:-
25 = 5^2
121 = 11^2
125 = (5)^1 + 2 = 5 ^3
126 = 21 x 6
127 = -1 + 2^7
289 = (8 + 9)^2
343 = (3 + 4 ) ^3
Friedman numbers are named after Erich Friedman, an Associate Professor of Mathematics in Florida in US.
Enjoy
Rajesh Thakur
References:-
1. https://en.wikipedia.org/wiki/Friedman_number
2. http://www2.stetson.edu/~efriedma/mathmagic/0800.html
Monday, September 12, 2016
Monday, April 4, 2016
RAMANUJAN TRIPLES
A number that can be written as the sum of cubes of a number in three ways.
This is also known as TAXICAB 3
DR RAJESH KUMAR THAKUR
Ramanujan Number
Ramanujan was fond of numbers. Prof Hardy once visited the hospital to see the ailing Ramanujan riding on a taxi. The taxi number was 1729. This 1729 is called the Ramanujan Number.
C P Show in his book wrote -
“Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’”
a^ 3 + b^ 3 = x where x is the Ramanujan Number
This number is also known as TAXI CAB number.
Thanks for reading
Dr Rajesh Kumar Thakur
Thursday, March 24, 2016
Kaprekar number
Kaprekar Number
D R Kaprekar (1905 - 1986)
A Kaprekar number is a special n digit number such that if it is squared the sum of the squared quantity’s right most n digits and remaining part are equal to the number itself.
45^2 = 2025 and 20 + 25 = 45
703^2 = 494 209 and 494 + 209 = 703
99^2 = 9801 and 98 + 01 = 99
The first few Kaprekar numbers are -
1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643,-----
The Kaprekar numbers were named after Shri Dattathreya Ramchandra Kaprekar who discovered them.
Dr Rajesh Kumar Thakur
A Kaprekar number is a special n digit number such that if it is squared the sum of the squared quantity’s right most n digits and remaining part are equal to the number itself.
45^2 = 2025 and 20 + 25 = 45
703^2 = 494 209 and 494 + 209 = 703
99^2 = 9801 and 98 + 01 = 99
The first few Kaprekar numbers are -
1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643,-----
The Kaprekar numbers were named after Shri Dattathreya Ramchandra Kaprekar who discovered them.
Dr Rajesh Kumar Thakur
Truncating Primes
Truncating Primes
There are few primes which remain Primes after chopping off their last digits.
73939133
7393913
739391
73939
7393
739
73
7
Let’s enjoy another
31
331
3331
33331
333331
3333331
33333331
Besides that 58393339, 37337999 …. are truncated Primes.
Dr Rajesh Kumar Thakur
There are few primes which remain Primes after chopping off their last digits.
73939133
7393913
739391
73939
7393
739
73
7
Let’s enjoy another
31
331
3331
33331
333331
3333331
33333331
Besides that 58393339, 37337999 …. are truncated Primes.
Dr Rajesh Kumar Thakur
Sunday, March 20, 2016
Palindrome Number
Palindromic Number:- Numbers which remains the same when read from the left to right or vice-versa. Indian mathematician Mahavira has mentioned about this in his book.
e.g. 15151, 11000011000011.
If you want to make any number Palindromic here is the simple way :-
Take a number 19
Reverse its digits + 91
Sum it 110
Reverse the digit + 011
Sum it 121
The process will be repeated until you get a palindromic number.
There are several Palindromes number. Mahavira called it पुष्प माल संख्या (garland number).
(Read about Palindrome Primes at -http://www.magic-squares.net/primes.htm)
There are several magic squares formed with the help of Palindromes number.
(Acknowledgement :- I acknowledge the effort by the above mentioned websites for their contribution in the field of Mathematics.I have tried to put their websites for further reference)
Dr Rajesh Kumar Thakru
e.g. 15151, 11000011000011.
If you want to make any number Palindromic here is the simple way :-
Take a number 19
Reverse its digits + 91
Sum it 110
Reverse the digit + 011
Sum it 121
The process will be repeated until you get a palindromic number.
There are several Palindromes number. Mahavira called it पुष्प माल संख्या (garland number).
(Read about Palindrome Primes at -http://www.magic-squares.net/primes.htm)
There are several magic squares formed with the help of Palindromes number.
(Acknowledgement :- I acknowledge the effort by the above mentioned websites for their contribution in the field of Mathematics.I have tried to put their websites for further reference)
Dr Rajesh Kumar Thakru
Friend Or Amicable Numbers
Amicable /Friend Numbers
The set of 220 and 284 was the first known set of amicable numbers. Pythagoras discovered the relationship and coined the term Amicable because he considered the numbers to be a symbol of friendship.
There is an unauthenticated story related to Amicable numbers that a prince whose name from the stand point of numerology was equivalent to 284 sought a bride whose name would represent 220 believing the fact that such combination would guarantee for a happy marriage throughout the life.
Here is the list of first ten Amicable Pairs:-
1 --- (220 ,284)
2 --- (1184 , 1210)
3 --- (2620 , 2924)
4 --- (5020 ,5564 )
5 --- (6232 ,6368)
6 --- (10744 ,10856)
7 -- (12285, 14595)
8 -- (17296 , 18416)
9 -- (63020 , 76048)
10 -- (66928 , 66992)
Dr Rajesh Kumar Thakur
Two numbers are called Amicable if each of which is equal to the sum of all the exact divisors of the other except the number itself. 220 and 284 are the first pair of amicable numbers.
220 has the exact divisors 1,2,4,5,10,11,20,22,44,55 and 110 whose sum is 284
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
and 284 has the exact divisors 1, 2, 4, 71, and 142 whose sum is 220.
1 + 2 + 4 + 71 + 142 = 220
Another pair of amicable number is 1184 and 1210.
There is an unauthenticated story related to Amicable numbers that a prince whose name from the stand point of numerology was equivalent to 284 sought a bride whose name would represent 220 believing the fact that such combination would guarantee for a happy marriage throughout the life.
1 --- (220 ,284)
2 --- (1184 , 1210)
3 --- (2620 , 2924)
4 --- (5020 ,5564 )
5 --- (6232 ,6368)
6 --- (10744 ,10856)
7 -- (12285, 14595)
8 -- (17296 , 18416)
9 -- (63020 , 76048)
10 -- (66928 , 66992)
An Arab Mathematician tried to generalize a rule to find the Amicable Number but he could not succed completely. The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the ninth century by the Arab mathematician Thābit ibn Qurra.
It states that if
where n > 1 is an integer and p, q, and r are prime numbers, then 2^n×p×q and 2^n ×r are a pair of amicable numbers.
Dr Rajesh Kumar Thakur
Beast Number
Beast Number:-
666 is called the Beast number. This is termed as unpleasant number. The direct reference of the number 666 can be found in the last book of Bible ‘Revelation’ in chapter 13, verse 18.
It can be written in the following ways:-
666 = 1 + 2 + 3 + 4 + 567 + 89
666 = 123 + 456 + 78 + 9
666 = 9 + 87 + 6 + 543 + 21
In war times this number becomes the handy tools in the hands of the propagandists to ascribe these letters to their opponents by assigning suitable letters pf the alphabet. During Second World War Hitler was ascribed with the title of beast by assigning each letter of English alphabet to a consecutive whole number beginning from 100 for A.
H 107
I 108
T 119
L 111
E 104
R 117
HITLER = 666
The beast number can be written with the help of sixth power of first three natural numbers
666 = 1^6 - 2^6 + 3^6
It can be written as the sum of squares of first seven primes
Dr Rajesh Kumar Thakur
666 is called the Beast number. This is termed as unpleasant number. The direct reference of the number 666 can be found in the last book of Bible ‘Revelation’ in chapter 13, verse 18.
It can be written in the following ways:-
666 = 1 + 2 + 3 + 4 + 567 + 89
666 = 123 + 456 + 78 + 9
666 = 9 + 87 + 6 + 543 + 21
In war times this number becomes the handy tools in the hands of the propagandists to ascribe these letters to their opponents by assigning suitable letters pf the alphabet. During Second World War Hitler was ascribed with the title of beast by assigning each letter of English alphabet to a consecutive whole number beginning from 100 for A.
H 107
I 108
T 119
L 111
E 104
R 117
HITLER = 666
The beast number can be written with the help of sixth power of first three natural numbers
666 = 1^6 - 2^6 + 3^6
It can be written as the sum of squares of first seven primes
Dr Rajesh Kumar Thakur
Prime Number
Prime number-
An integer p which is not 0 or ± 1 and is divisible by no integer except ±1 and itself is called Prime number. Donzager stated “Upon looking at prime numbers one has the feeling of being in presence of one the inexplicable phenomena one site of creation.”
2 is the only prime number. There is no perfect technique which can tell us immediately the numbers of prime between two numbers. Though Erastothenes, a great Greek mathematician suggested a method to find the primes between two numbers called Sieve of Erastothenes.
Erastothenes(276-195BC) gave a golden rule though simple it is time consuming which states “First write down the number from 2 to N. Remove all the multiples of 2, 3 and continue the process until all the multiple of primes not greater than √N has been removed.”
Suppose we have to find the primes below 30, first we find the square root of 30 which is 5.477.so we need to remove the entire multiple up to primes 5.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Hence the primes below 30 are 2,3,5,7,11,13,17,19,23 and,29.
Properties of Primes:-
i) Every natural number greater than 1 has at least one prime.
ii) For every integer a, a^p –a is always divisible by prime p
p 2 2 2 2 3 3 3 3 5 5 5 5
a 2 3 4 5 2 3 4 5 2 3 4 5
a^p-a 2 6 12 20 6 24 60 120 30 240 1020 3120
iii) If N is a prime number then 1+ (N-1)! Is always divisible by N. This is called Wilson theorem.
For N = 2, 1+ (2-1)! = 2 us divisible by 2.
iv) Every odd integers is the sum of a prime and a power of 2, this was claimed in 1848 by De Polignac.
55 = 47+23
v) Every even number except 2 is the sum of two prime numbers.
e.g. 8 = 3 + 5, 16 = 13 + 3 ,60 = 13 + 47 etc. This is called Goldbach conjecture.
vi) Every even integer greater than 4 can be written as the sum of two odd prime numbers. 4 = 2 + 2 = 1 + 3, 6 = 3 + 3 = 1 + 5 …
vii) There is always at least one prime number between n and 2n-2 provided n is greater than 3. If n = 4, 2n-2 = 6 then obviously 5 lies in between 4 and 6. This conjecture was stated by Bertrand (1822-1903).
Interesting Facts
1. A pair of prime numbers is said to be a twin prime pair if the two numbers differ by any 2.
i.e. (3,5)(5,7)(11,13)(17,19)(29,31)(41,43)(59,61)(71,73)etc.
All the twin primes are of the form 6n-1, 6n+1.
2. Between 9,999,900 to 10, 000, 000, there are only 9 prime numbers.
9,999,901; 9,999,907; 9,999,929; 9,999,931 ;9,999,937 ;9,999,943 ;9,999,971 ;9,999,973 ;9,999,991.
But in the next 100 integer from 10,000,000 to 10, 000,100 there are only two primes 10, 000, 019 and 10,000,079.
3. (p,p+2,p+4)is called prime triplet if all numbers are primes
4. The largest known prime number is of 6, 320, 430, digits and was found by Michael Shafer in Dec 2003. It would need 1400 to 1500 pages to write.
5. A gap of 803 composite numbers exits between primes 90874329411493 and 90874329412297 which was found in 1989 by J.Yong and A.Poster
Dr. Rajesh Kumar Thakur
An integer p which is not 0 or ± 1 and is divisible by no integer except ±1 and itself is called Prime number. Donzager stated “Upon looking at prime numbers one has the feeling of being in presence of one the inexplicable phenomena one site of creation.”
2 is the only prime number. There is no perfect technique which can tell us immediately the numbers of prime between two numbers. Though Erastothenes, a great Greek mathematician suggested a method to find the primes between two numbers called Sieve of Erastothenes.
Erastothenes(276-195BC) gave a golden rule though simple it is time consuming which states “First write down the number from 2 to N. Remove all the multiples of 2, 3 and continue the process until all the multiple of primes not greater than √N has been removed.”
Suppose we have to find the primes below 30, first we find the square root of 30 which is 5.477.so we need to remove the entire multiple up to primes 5.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Hence the primes below 30 are 2,3,5,7,11,13,17,19,23 and,29.
Properties of Primes:-
i) Every natural number greater than 1 has at least one prime.
ii) For every integer a, a^p –a is always divisible by prime p
p 2 2 2 2 3 3 3 3 5 5 5 5
a 2 3 4 5 2 3 4 5 2 3 4 5
a^p-a 2 6 12 20 6 24 60 120 30 240 1020 3120
iii) If N is a prime number then 1+ (N-1)! Is always divisible by N. This is called Wilson theorem.
For N = 2, 1+ (2-1)! = 2 us divisible by 2.
iv) Every odd integers is the sum of a prime and a power of 2, this was claimed in 1848 by De Polignac.
55 = 47+23
v) Every even number except 2 is the sum of two prime numbers.
e.g. 8 = 3 + 5, 16 = 13 + 3 ,60 = 13 + 47 etc. This is called Goldbach conjecture.
vi) Every even integer greater than 4 can be written as the sum of two odd prime numbers. 4 = 2 + 2 = 1 + 3, 6 = 3 + 3 = 1 + 5 …
vii) There is always at least one prime number between n and 2n-2 provided n is greater than 3. If n = 4, 2n-2 = 6 then obviously 5 lies in between 4 and 6. This conjecture was stated by Bertrand (1822-1903).
Interesting Facts
1. A pair of prime numbers is said to be a twin prime pair if the two numbers differ by any 2.
i.e. (3,5)(5,7)(11,13)(17,19)(29,31)(41,43)(59,61)(71,73)etc.
All the twin primes are of the form 6n-1, 6n+1.
2. Between 9,999,900 to 10, 000, 000, there are only 9 prime numbers.
9,999,901; 9,999,907; 9,999,929; 9,999,931 ;9,999,937 ;9,999,943 ;9,999,971 ;9,999,973 ;9,999,991.
But in the next 100 integer from 10,000,000 to 10, 000,100 there are only two primes 10, 000, 019 and 10,000,079.
3. (p,p+2,p+4)is called prime triplet if all numbers are primes
4. The largest known prime number is of 6, 320, 430, digits and was found by Michael Shafer in Dec 2003. It would need 1400 to 1500 pages to write.
5. A gap of 803 composite numbers exits between primes 90874329411493 and 90874329412297 which was found in 1989 by J.Yong and A.Poster
Dr. Rajesh Kumar Thakur
Saturday, February 27, 2016
Mathematically 100
One Hundred (100)
1. It is the smallest three digit number.
2. It is the sum of first 10 odd numbers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
3. Boiling point of water is 100^0C
4. It is the sum of 2 consecutive triangular numbers.
45 + 55 = 100
5. It is the sum of 9 consecutive primes.
2 + 3 + 5 + 7 + 11 +13 +17 + 19 + 23 = 100
6. It is a square number.
10^2 = 100
7. It can be expressed as the sum of cube of first four natural numbers.
1^3 + 2^3 + 3^3 + 4^3 = 100
8. It is an abundant number.
9. It is a centered 33 gonal number.
10. It can be expressed as the sum of four pairs of prime numbers.
47 + 53 = 17 + 83 = 3 + 97 = 41 + 59 = 100
11. It is a Harshad number in base 10.
12. It is a Leyland number which can be expressed in the form of a^b + b^a .
100 = 2^6 + 6^2
13. It is the smallest number whose common logarithm is a prime number.
log 10^100 = 2
14. The atomic number of Fermium is 100.
15. A century has 100 years.
16. There is a very interesting puzzle which says – using digits from 1 to 9, and mathematical operators + and - , make 100. There are several solutions to this puzzle few of them are provided here.
98 – 76 + 54 + 3 + 21 =100
9 – 8 + 76 + 54 – 32 + 1 = 100
98 + 7 + 6 – 5 – 4 – 3 + 2 – 1 = 100
9 – 8 + 76 – 5 + 4 + 3 + 21 = 100
98 – 7 – 6 – 5 – 4 + 3 + 21 = 100
Send Your Valuable Comments on
rkthakur1974@gmail.com
Dr Rajesh Kumar Thakur
1. It is the smallest three digit number.
2. It is the sum of first 10 odd numbers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100
3. Boiling point of water is 100^0C
4. It is the sum of 2 consecutive triangular numbers.
45 + 55 = 100
5. It is the sum of 9 consecutive primes.
2 + 3 + 5 + 7 + 11 +13 +17 + 19 + 23 = 100
6. It is a square number.
10^2 = 100
7. It can be expressed as the sum of cube of first four natural numbers.
1^3 + 2^3 + 3^3 + 4^3 = 100
8. It is an abundant number.
9. It is a centered 33 gonal number.
10. It can be expressed as the sum of four pairs of prime numbers.
47 + 53 = 17 + 83 = 3 + 97 = 41 + 59 = 100
11. It is a Harshad number in base 10.
12. It is a Leyland number which can be expressed in the form of a^b + b^a .
100 = 2^6 + 6^2
13. It is the smallest number whose common logarithm is a prime number.
log 10^100 = 2
14. The atomic number of Fermium is 100.
15. A century has 100 years.
16. There is a very interesting puzzle which says – using digits from 1 to 9, and mathematical operators + and - , make 100. There are several solutions to this puzzle few of them are provided here.
98 – 76 + 54 + 3 + 21 =100
9 – 8 + 76 + 54 – 32 + 1 = 100
98 + 7 + 6 – 5 – 4 – 3 + 2 – 1 = 100
9 – 8 + 76 – 5 + 4 + 3 + 21 = 100
98 – 7 – 6 – 5 – 4 + 3 + 21 = 100
Send Your Valuable Comments on
rkthakur1974@gmail.com
Dr Rajesh Kumar Thakur
Mathematically 98
Ninety Eight (98)
1. It can be expressed as the fourth power of sum of first three natural numbers.
98 = 1^4 + 2^4 + 3^4
2. The atomic number of Californium is 98.
3. It can be expressed as the difference between two odd consecutive cubes.
98 = 53 – 33
4. The reciprocal of 98 = 1/98 on division shows an interesting pattern of GP, though the patterns discontinues but it starts with the power of 2.
1/98 = 0. 01 02 04 08 16 32 65 30 ….
Here the number shown in bold is in GP.
5. It is the lowest number such that the first 5 multiple of it contains the digit 9.
98 x 1 = 98
98 x 2 = 196
98 x 3 = 294
98 x 4 = 392
98 x 5 = 490
6. It is the smallest number which cannot be expressed as the sum of two primes if the first prime is either of 3, 5 or 7. The next such numbers are 122, 124, 126 and 128.
Dr Rajesh Kumar Thakur
1. It can be expressed as the fourth power of sum of first three natural numbers.
98 = 1^4 + 2^4 + 3^4
2. The atomic number of Californium is 98.
3. It can be expressed as the difference between two odd consecutive cubes.
98 = 53 – 33
4. The reciprocal of 98 = 1/98 on division shows an interesting pattern of GP, though the patterns discontinues but it starts with the power of 2.
1/98 = 0. 01 02 04 08 16 32 65 30 ….
Here the number shown in bold is in GP.
5. It is the lowest number such that the first 5 multiple of it contains the digit 9.
98 x 1 = 98
98 x 2 = 196
98 x 3 = 294
98 x 4 = 392
98 x 5 = 490
6. It is the smallest number which cannot be expressed as the sum of two primes if the first prime is either of 3, 5 or 7. The next such numbers are 122, 124, 126 and 128.
Dr Rajesh Kumar Thakur
Mathematically 97
Ninety Seven (97)
1. It is a 25th prime number and the largest two digit prime number.
2. It is the highest two digit prime number whose cube has no zero.
97^3 = 912673
3. In Georgian calendar there are 97 leap days in every 400 years.
4. It is a 4 dimensional centered cube number.
5. It is a permutable primes. A prime is called permutable if on a given base, the prime always remain a prime despite its digit’s position changed through permutation. It was first studied by H E Richert. 97 and its reverse 79 are primes .The first 10 permutable primes are- 2, 3, 5, 7, 11, 13, 17, 31, 37 and 71.
6. It can be written in the form of – n^4 + (n +1)^4
97 = 2^4 + 3^4
7. The period of reciprocal of 97 = 1/97 is maximum and its length is 96. More interestingly, the first four pairs of its expansion are in Geometric Progression (GP). Alexender Aitken, a professor at Edinburgh University knew it by heart.
1/97 = 0.01 03 09 27 83 50 51 57 52 57 73 19 58 76 28 86 59 79……
8. It is a Proth prime. A Proth prime is written in the form of – k x 2n + 1, where k is an odd positive integer and 2n > k. It is named after the mathematician Francois Proth. The first 10 Proth primes are- 3, 5, 9, 13, 17, 25, 33, 41, 49 and 57.
97 = 3 x 25 + 1
9. The atomic number of Berkelium is 97
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a 25th prime number and the largest two digit prime number.
2. It is the highest two digit prime number whose cube has no zero.
97^3 = 912673
3. In Georgian calendar there are 97 leap days in every 400 years.
4. It is a 4 dimensional centered cube number.
5. It is a permutable primes. A prime is called permutable if on a given base, the prime always remain a prime despite its digit’s position changed through permutation. It was first studied by H E Richert. 97 and its reverse 79 are primes .The first 10 permutable primes are- 2, 3, 5, 7, 11, 13, 17, 31, 37 and 71.
6. It can be written in the form of – n^4 + (n +1)^4
97 = 2^4 + 3^4
7. The period of reciprocal of 97 = 1/97 is maximum and its length is 96. More interestingly, the first four pairs of its expansion are in Geometric Progression (GP). Alexender Aitken, a professor at Edinburgh University knew it by heart.
1/97 = 0.01 03 09 27 83 50 51 57 52 57 73 19 58 76 28 86 59 79……
8. It is a Proth prime. A Proth prime is written in the form of – k x 2n + 1, where k is an odd positive integer and 2n > k. It is named after the mathematician Francois Proth. The first 10 Proth primes are- 3, 5, 9, 13, 17, 25, 33, 41, 49 and 57.
97 = 3 x 25 + 1
9. The atomic number of Berkelium is 97
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 96
Ninety Six (96)
1. It is a sixth star number.
2. It is the smallest number that can be written as the difference of 2 squares in 4 ways.
96 = 252 – 232 = 142 - 102 = 112 – 52 = 102 – 22
3. It can be expressed as the difference of factorials of two consecutive numbers.
96 = 5! – 4! = 120 – 24
4. It is the 7th number that stays same when written upside down.
5. It is the second smallest number with 6 prime factors.
96 = 3 x 2 x 2 x 2 x 2 x 2
6. It is a refactorable number. A number is called refactorable or tau number if it is divisible by the count of its divisors. 96 has 12 divisors – 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96. Such numbers were first studied by Curtis Cooper and Robert E Kennedy and later rediscovered by Simon Colton. The first 10 Tau numbers are- 1, 2, 8, 9, 12, 18, 24, 36, 40 and 56.
7. It is an octagonal number.
8. It is an abundant number.
9. It is a 33 gonal number.
10. The atomic number of Curium is 96.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a sixth star number.
2. It is the smallest number that can be written as the difference of 2 squares in 4 ways.
96 = 252 – 232 = 142 - 102 = 112 – 52 = 102 – 22
3. It can be expressed as the difference of factorials of two consecutive numbers.
96 = 5! – 4! = 120 – 24
4. It is the 7th number that stays same when written upside down.
5. It is the second smallest number with 6 prime factors.
96 = 3 x 2 x 2 x 2 x 2 x 2
6. It is a refactorable number. A number is called refactorable or tau number if it is divisible by the count of its divisors. 96 has 12 divisors – 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96. Such numbers were first studied by Curtis Cooper and Robert E Kennedy and later rediscovered by Simon Colton. The first 10 Tau numbers are- 1, 2, 8, 9, 12, 18, 24, 36, 40 and 56.
7. It is an octagonal number.
8. It is an abundant number.
9. It is a 33 gonal number.
10. The atomic number of Curium is 96.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 95
Ninety Five (95)
1. It is the sum of 7 consecutive prime starting from 5.
95 = 5 + 7 + 11 + 13 + 17 + 19 + 23
2. It can be expressed as the sum of squares of 4 numbers distinctly.
95 = 1^2 + 2^2 + 3^2 + 9^2
3. It is a hexagonal pyramidal number.
4. It is a Thabit number. A number that can be expresses as 3 x 2n – 1 for n > 0 is known as Thabit number. It is named after Iraqi mathematician of 9th century Thabit ibn Qurra. The first ten Thabit numbers are- 2, 5, 11, 23, 47, 95, 191, 383, 767 and 1535.
5. The number of planar partition of 10 is 95.
6. The atomic number of Americium is 95.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the sum of 7 consecutive prime starting from 5.
95 = 5 + 7 + 11 + 13 + 17 + 19 + 23
2. It can be expressed as the sum of squares of 4 numbers distinctly.
95 = 1^2 + 2^2 + 3^2 + 9^2
3. It is a hexagonal pyramidal number.
4. It is a Thabit number. A number that can be expresses as 3 x 2n – 1 for n > 0 is known as Thabit number. It is named after Iraqi mathematician of 9th century Thabit ibn Qurra. The first ten Thabit numbers are- 2, 5, 11, 23, 47, 95, 191, 383, 767 and 1535.
5. The number of planar partition of 10 is 95.
6. The atomic number of Americium is 95.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 94
Ninety Four (94)
1. It is a Smith number. A number is called Smith number if the sum of digits of its prime factor is equal to the sum of the digits of the number.
94 = 2 x 47 and 9 + 4 = 2 + 4 + 7
2. It is a 17- gonal number.
3. Apart from 2 and 4, it is the smallest even number which cannot be written as the sum of two distinct semi primes.
4. It is a centered 31 gonal number.
5. It can be partitioned 48 times with each term not larger than 2 and 784 times with each term not larger than 3.
6. The atomic number of Plutonium is 94. It is the notorious chemical element which doesn’t exist naturally and was first created artificially in 1940.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a Smith number. A number is called Smith number if the sum of digits of its prime factor is equal to the sum of the digits of the number.
94 = 2 x 47 and 9 + 4 = 2 + 4 + 7
2. It is a 17- gonal number.
3. Apart from 2 and 4, it is the smallest even number which cannot be written as the sum of two distinct semi primes.
4. It is a centered 31 gonal number.
5. It can be partitioned 48 times with each term not larger than 2 and 784 times with each term not larger than 3.
6. The atomic number of Plutonium is 94. It is the notorious chemical element which doesn’t exist naturally and was first created artificially in 1940.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 93
Ninety Three (93)
1. With just a straight cut, a potato can be divided into 93 pieces.
2. It is an odd number with the property that the digit at ten’s place is thrice the digit at one’s place.
3. It is a 32 gonal number.
4. It can be partitioned 47 times with each tern not larger than 2 and 768 times where each term is not larger than 3.
5. It is the first member of the triplet of consecutive semi-primes 93, 94 and 95. A number is called semi primes if it can be expressed as the product of two prime numbers.
93 = 3 x 31 94 = 2 x 47 95 = 5 x 19
6. It is a palindrome number on base 2.
(93)10 = (1011101)2
7. On base 5, it is a repeated digit.
(93)10 = (333)5
8. It can be expressed as the sum of squares of three numbers. Interestingly all the three numbers used are in Arithmetic Progression with common difference 3.
93 = 2^2 + 5^2 + 8^2
9. It is the sum of 6 consecutive numbers.
93 = 13 + 14 + 15 + 16 + 17 + 18
10. It is a Blum integer. A natural number n is a Blum integer if n = p x q is a semi prime and p and q are written in the form of 4x + 3. It is named after computer scientist Manuel Blum.
93 = 3 x 31
where 3 and 31 are prime numbers
31 = 4 x 7 + 3
11. The atomic number of Neptunium is 93.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. With just a straight cut, a potato can be divided into 93 pieces.
2. It is an odd number with the property that the digit at ten’s place is thrice the digit at one’s place.
3. It is a 32 gonal number.
4. It can be partitioned 47 times with each tern not larger than 2 and 768 times where each term is not larger than 3.
5. It is the first member of the triplet of consecutive semi-primes 93, 94 and 95. A number is called semi primes if it can be expressed as the product of two prime numbers.
93 = 3 x 31 94 = 2 x 47 95 = 5 x 19
6. It is a palindrome number on base 2.
(93)10 = (1011101)2
7. On base 5, it is a repeated digit.
(93)10 = (333)5
8. It can be expressed as the sum of squares of three numbers. Interestingly all the three numbers used are in Arithmetic Progression with common difference 3.
93 = 2^2 + 5^2 + 8^2
9. It is the sum of 6 consecutive numbers.
93 = 13 + 14 + 15 + 16 + 17 + 18
10. It is a Blum integer. A natural number n is a Blum integer if n = p x q is a semi prime and p and q are written in the form of 4x + 3. It is named after computer scientist Manuel Blum.
93 = 3 x 31
where 3 and 31 are prime numbers
31 = 4 x 7 + 3
11. The atomic number of Neptunium is 93.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 92
Ninety Two (92)
1. It is an eighth pentagonal number.
2. It is the maximal number of regions into which 13 lines divide a plane.
3. The maximum number of regions into which a plane can be divided by 10 circles is 92.
4. A snub dodecahedron is a solid with 92 faces,, each consisting of 80 equilateral triangle and 12 pentagons.
5. There is a very interesting puzzle involving 92.
Puzzle: - Multiply 92 by 8, then the product by 8 and so on. List the product one under the other shifting the digit two places to the right and add, the sum of these number is a string of 9.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is an eighth pentagonal number.
2. It is the maximal number of regions into which 13 lines divide a plane.
3. The maximum number of regions into which a plane can be divided by 10 circles is 92.
4. A snub dodecahedron is a solid with 92 faces,, each consisting of 80 equilateral triangle and 12 pentagons.
5. There is a very interesting puzzle involving 92.
Puzzle: - Multiply 92 by 8, then the product by 8 and so on. List the product one under the other shifting the digit two places to the right and add, the sum of these number is a string of 9.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 91
Ninety One (91)
1. It is the sum of first 13 consecutive number.
1 + 2 + 3 +--- + 13 = 91
2. The sum of consecutive squares of first 6 numbers is 91.
1^2 + 2^2 + 3^2 +4^2 +5^2 + 6^2 = 91
3. A 6 x 6 square board has 91 square in total.
4. It is a centered cube number.
5. It is a pyramidal number.
6. It is the sum of cubes of two consecutive numbers.
91 = 3^3 + 4^3
7. It is a centered 15 gonal and 30 gonal numbers.
8. The number of days in a quarter year is 91. A year has 52 weeks so a quarter year has 13 weeks, each of 7 days.
9. It is a centered hexagonal number.
1 + 6 + 12 + 18 + 24 + 30 = 91
10. It is the smallest positive integer which can be expressed as the sum of distinct squares in different ways.
91 = 1^2 + 3^2 + 9^2
= 1^2 + 4^2 + 5^2 + 7^2
11. It is a repeated digit on base 9.
91 = (111)9
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the sum of first 13 consecutive number.
1 + 2 + 3 +--- + 13 = 91
2. The sum of consecutive squares of first 6 numbers is 91.
1^2 + 2^2 + 3^2 +4^2 +5^2 + 6^2 = 91
3. A 6 x 6 square board has 91 square in total.
4. It is a centered cube number.
5. It is a pyramidal number.
6. It is the sum of cubes of two consecutive numbers.
91 = 3^3 + 4^3
7. It is a centered 15 gonal and 30 gonal numbers.
8. The number of days in a quarter year is 91. A year has 52 weeks so a quarter year has 13 weeks, each of 7 days.
9. It is a centered hexagonal number.
1 + 6 + 12 + 18 + 24 + 30 = 91
10. It is the smallest positive integer which can be expressed as the sum of distinct squares in different ways.
91 = 1^2 + 3^2 + 9^2
= 1^2 + 4^2 + 5^2 + 7^2
11. It is a repeated digit on base 9.
91 = (111)9
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Tuesday, February 9, 2016
Mathematically 90
Ninety (90)
1. It is the product of two consecutive integers. Such numbers are called Oblong numbers because it corresponds to the area of an oblong rectangle whose length is one greater than its width.
90 = 9 x 10
2. An angle of 90 degree is called a right angle.
3. Tangents touches the circle at 90 degree.
4. It is the sum of first 9 consecutive even numbers
. 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = 90
5. It is the sum of two successive primes.
90 = 43 + 47
6. The interior angle of a square is 90 degree.
7. It is a Harshad number because it is divisible by sum of its digits.
8. The atomic number of Thorium is 90.
9. It is a 31 gonal number.
10. A truncated dodecahedron had 90 edges.
Dr Rajesh Kr Thakur
rkthakur1974@gmail.com
1. It is the product of two consecutive integers. Such numbers are called Oblong numbers because it corresponds to the area of an oblong rectangle whose length is one greater than its width.
90 = 9 x 10
2. An angle of 90 degree is called a right angle.
3. Tangents touches the circle at 90 degree.
4. It is the sum of first 9 consecutive even numbers
. 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 = 90
5. It is the sum of two successive primes.
90 = 43 + 47
6. The interior angle of a square is 90 degree.
7. It is a Harshad number because it is divisible by sum of its digits.
8. The atomic number of Thorium is 90.
9. It is a 31 gonal number.
10. A truncated dodecahedron had 90 edges.
Dr Rajesh Kr Thakur
rkthakur1974@gmail.com
Mathematically 89
Eighty Nine (89)
1. It is a Fibonacci prime number.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
2. The inverse of 89 gives a fraction with a sequence of 44 recurring digits.
1/89 = 0. 01123595505617977528089887640449438202247191…
The best part of the result is that this number is the sum of the sequence of number from the Fibonacci series where by each number moves a decimal point to the right.
0.01 + 0.001 + 0.0002 + 0.00003 +0.000005 +0.0000008 +0.00000013 +0.000000021 + 0.0000000034 +…=0.011235955---
3. The first pair of consecutive prime that differ by 8 is 89 and 97.
4. It is a Markov number discovered by Andrey Markoff. A Markov number is a positive integer x, y or z such that-
x^3 +y^3+z^3 = 3xyz
The first ten Markov numbers are 1, 2, 5, 13, 29, 34, 89, 169, 194 and 233.
5. It is a Chen prime named after Chen Jingrun. A number p is a Chen prime if p+2 is either a prime or a product of two primes. The first ten Chen primes are- 2, , 5, 7, 11, 13, 17, 19, 23 and 29.
6. The atomic number of Actinium is 89.
Dr Rajesh K Thakur
rkthakur1974@gmail.com
1. It is a Fibonacci prime number.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
2. The inverse of 89 gives a fraction with a sequence of 44 recurring digits.
1/89 = 0. 01123595505617977528089887640449438202247191…
The best part of the result is that this number is the sum of the sequence of number from the Fibonacci series where by each number moves a decimal point to the right.
0.01 + 0.001 + 0.0002 + 0.00003 +0.000005 +0.0000008 +0.00000013 +0.000000021 + 0.0000000034 +…=0.011235955---
3. The first pair of consecutive prime that differ by 8 is 89 and 97.
4. It is a Markov number discovered by Andrey Markoff. A Markov number is a positive integer x, y or z such that-
x^3 +y^3+z^3 = 3xyz
The first ten Markov numbers are 1, 2, 5, 13, 29, 34, 89, 169, 194 and 233.
5. It is a Chen prime named after Chen Jingrun. A number p is a Chen prime if p+2 is either a prime or a product of two primes. The first ten Chen primes are- 2, , 5, 7, 11, 13, 17, 19, 23 and 29.
6. The atomic number of Actinium is 89.
Dr Rajesh K Thakur
rkthakur1974@gmail.com
Mathematically 88
Eighty Eight (88)
1. A piano has 88 keys with 36 black and 52 white.
2. It is the 6th number that stays same when written upside down.
3. It is the only known number whose square has no isolated digits.
88^2 = 7744
4. It is the sum of 4 consecutive primes.
88 = 17 + 19 + 23 + 29
5. It is an untouchable number. An untouchable number is a positive integer that can’t be expressed as the sum of all the proper divisor of any number.
The first 10 untouchable numbers are- 2, 5, 52, 88, 96, 120, 124, 146, 162 and 188.
6. International Astronomical Union has defined 88 constellations divided into 8 families—Ursa Major family, Zodiacal family, Perseus family, Heavenly waters, Orion Group, Bayer Group, La Cille family and Hercules family.
7. It is a 16 gonal number.
8. It is an abundant number.
9. It is a centered 29-gonal number.
Dr Rajesh K Thakur
rkthakur1974@gmail.com
1. A piano has 88 keys with 36 black and 52 white.
2. It is the 6th number that stays same when written upside down.
3. It is the only known number whose square has no isolated digits.
88^2 = 7744
4. It is the sum of 4 consecutive primes.
88 = 17 + 19 + 23 + 29
5. It is an untouchable number. An untouchable number is a positive integer that can’t be expressed as the sum of all the proper divisor of any number.
The first 10 untouchable numbers are- 2, 5, 52, 88, 96, 120, 124, 146, 162 and 188.
6. International Astronomical Union has defined 88 constellations divided into 8 families—Ursa Major family, Zodiacal family, Perseus family, Heavenly waters, Orion Group, Bayer Group, La Cille family and Hercules family.
7. It is a 16 gonal number.
8. It is an abundant number.
9. It is a centered 29-gonal number.
Dr Rajesh K Thakur
rkthakur1974@gmail.com
Mathematically 87
Eighty Seven (87)
Number
Divisors
1. It is the sum of divisors of the first ten integers.
2. It is the sum of square of the first four primes.
87 = 2^2 + 3^2 + 5^2 + 7^2
3. It is the sum of 6 consecutive numbers.
87 = 12 + 13 + 14 + 15 + 16 + 17
4. It is a 30 –gonal number.
5. It can be partitioned 44 times with each term not larger than 2
6. It is regarded as an unlucky number in Cricket as it is 13 short of 100.
Dr Rajesh Kr Thakur
rkthakur1974@gmail.com
Number
1. It is the sum of divisors of the first ten integers.
2. It is the sum of square of the first four primes.
87 = 2^2 + 3^2 + 5^2 + 7^2
3. It is the sum of 6 consecutive numbers.
87 = 12 + 13 + 14 + 15 + 16 + 17
4. It is a 30 –gonal number.
5. It can be partitioned 44 times with each term not larger than 2
6. It is regarded as an unlucky number in Cricket as it is 13 short of 100.
Dr Rajesh Kr Thakur
rkthakur1974@gmail.com
Mathematically 86
Eighty Six (86)
1. It is the largest known n for which 2^n contains no zeros.
2^86 = 77, 371,252,455,336,267,181,195,264
2. It is the sum of 4 consecutive squares.
86 = 3^2 + 4^2 + 5^2 + 6^2
3. It is a member of a Padovan sequence. It is similar to Fibonacci sequence. The sequence runs from 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16 ,21, 28, 37, 49, 65, 86 …
The Padovan sequence is --A0 = 0; A1 = 1; A2 = 1; AN+1 = AN-1 + AN-2.
4. It is the total number of parts in all partitions of 8.
5. An ancient tablet was discovered with 86 notches on it. This tablet may also be a pregnancy calendar designed to estimate when a pregnant woman will give birth. This tablet is 32400 years old found in 1979 in Germany. On one side of the tablet is the man like being with his legs apart and arms raised. Between his legs hangs what could be a sword and his waist in narrow. His left leg is shorter than his right one. It is the number of days that must be subtracted from a year to equal the average number of days of a human gestation. It gives the roughly estimation that a child is born after 9 months.
365 – 86 = 279 = 9 x 30 + 9
6. It is a repeated digit on base 6.
86 = (222)6
7. It is the sum of four consecutive integers.
86= 20 + 21 + 22 + 23
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the largest known n for which 2^n contains no zeros.
2^86 = 77, 371,252,455,336,267,181,195,264
2. It is the sum of 4 consecutive squares.
86 = 3^2 + 4^2 + 5^2 + 6^2
3. It is a member of a Padovan sequence. It is similar to Fibonacci sequence. The sequence runs from 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16 ,21, 28, 37, 49, 65, 86 …
The Padovan sequence is --A0 = 0; A1 = 1; A2 = 1; AN+1 = AN-1 + AN-2.
4. It is the total number of parts in all partitions of 8.
5. An ancient tablet was discovered with 86 notches on it. This tablet may also be a pregnancy calendar designed to estimate when a pregnant woman will give birth. This tablet is 32400 years old found in 1979 in Germany. On one side of the tablet is the man like being with his legs apart and arms raised. Between his legs hangs what could be a sword and his waist in narrow. His left leg is shorter than his right one. It is the number of days that must be subtracted from a year to equal the average number of days of a human gestation. It gives the roughly estimation that a child is born after 9 months.
365 – 86 = 279 = 9 x 30 + 9
6. It is a repeated digit on base 6.
86 = (222)6
7. It is the sum of four consecutive integers.
86= 20 + 21 + 22 + 23
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 85
Eighty Five (85)
1. It can be expressed as the sum of two squares in two ways. The second pair is the sum of square of 2 consecutive numbers.
85 = 9^2 + 2^2
= 7^2 + 6^2
2. It is one of the member of a Pythagorean triplet (13, 84, 85).
85^2 = 13^2 + 84^2
3. It can be uniquely expresses as the sum of powers of 4 starting from 0 to 3.
85 = 4^0 + 4^1 +4^2 + 4^3
4. It can be partitioned 43 times with each term not larger than 3 and 645 times with each term not larger than 3
5. It is a decagonal number.
6. It is the product of two prime number and therefore is a bi-prime.
85 = 5 x 17
7. It is a centered triangular and centered square number. a centered square number is a figurative number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center in a successive layer.
The first few centered square numbers are- 1, 5, 13, 25…
8. On base 4 , it is a repeated unit number.
85 = (1111)4
9. It is a centered triangular number. a centered triagnualar number is a figurative number that represents a triangle with a dot in the center and other dots surrounding in successive triangular layer. The first few centered triangular numbers are—1, 4, 10, 19, 31….
10. It is the largest known n for which ---
1^2 + 2^2 + 3^2 +---- + n^2 = 1 + 2 +3 +---+m
Dr Rajesh Kr Thakur
rkthakur1974@gmail.com
1. It can be expressed as the sum of two squares in two ways. The second pair is the sum of square of 2 consecutive numbers.
85 = 9^2 + 2^2
= 7^2 + 6^2
2. It is one of the member of a Pythagorean triplet (13, 84, 85).
85^2 = 13^2 + 84^2
3. It can be uniquely expresses as the sum of powers of 4 starting from 0 to 3.
85 = 4^0 + 4^1 +4^2 + 4^3
4. It can be partitioned 43 times with each term not larger than 3 and 645 times with each term not larger than 3
5. It is a decagonal number.
6. It is the product of two prime number and therefore is a bi-prime.
85 = 5 x 17
7. It is a centered triangular and centered square number. a centered square number is a figurative number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center in a successive layer.
The first few centered square numbers are- 1, 5, 13, 25…
8. On base 4 , it is a repeated unit number.
85 = (1111)4
9. It is a centered triangular number. a centered triagnualar number is a figurative number that represents a triangle with a dot in the center and other dots surrounding in successive triangular layer. The first few centered triangular numbers are—1, 4, 10, 19, 31….
10. It is the largest known n for which ---
1^2 + 2^2 + 3^2 +---- + n^2 = 1 + 2 +3 +---+m
Dr Rajesh Kr Thakur
rkthakur1974@gmail.com
Monday, February 8, 2016
Mathematically 84
Eighty Four (84)
1. It is a tetrahedral number.
2. It is the sum of first 7 triangular number.
1 + 3 + 6 + 10 + 15 + 21 + 28 = 84
3. It is the sum of twin prime.
84 = 41 + 43
4. If you add 84 to 1000, the number 1084 you obtain is the smallest natural number which contains the five vowel in order.
1084 = One thousand eighty four
5. A hepteract is a seven dimensional hypercube with 84 penteract 5 faces. In geometry, a 7 cube or Hepteract is a seven dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4- faces, 84 penteract 5 faces, and 14 hexeract 6 faces.
6. It is an abundant number.
7. It is a 29-goanl number.
8. It can be partitioned 43 times with each term no longer than 2 and 631 times with each term no larger than 3.
9. It can be expressed as the sum of first 3 powers of 4 uniquely.
84 = 4^1 + 4^2 + 4^3
10. It is the maximal number of regions into which space can be divided by 7 spheres.
11. The probability of a pregnant woman having twins is 1 : 84.
12. According to Greek Anthalogy , Diophantus the Greek mathematician lived for 84 years. There is no other information known about his life except that comes from this problem which gives the result 84 on solving. The problem says---
This tomb holds Diophantus. Ah, how great a marvel! The tomb tells scientifically the measure of his life. God granted him to be a boy for 1/6 th of his life; and adding a twelfth part to this, he clothed his cheeks with down. He lit him the light of wedlock after a seventh part, and five years after his marriage he gave him a son. Alas, late born wretched child! After attaining the measure of half his father’s life, chill Fate took him. After consoling his grief by the study of numbers for four years, Diophantus ended his life. If Diophantus age is x years then we can write this algebraically as-- Which when solved gives x = 84. Diophanuts is better known as the father of algebra.
1. It is a tetrahedral number.
2. It is the sum of first 7 triangular number.
1 + 3 + 6 + 10 + 15 + 21 + 28 = 84
3. It is the sum of twin prime.
84 = 41 + 43
4. If you add 84 to 1000, the number 1084 you obtain is the smallest natural number which contains the five vowel in order.
1084 = One thousand eighty four
5. A hepteract is a seven dimensional hypercube with 84 penteract 5 faces. In geometry, a 7 cube or Hepteract is a seven dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4- faces, 84 penteract 5 faces, and 14 hexeract 6 faces.
6. It is an abundant number.
7. It is a 29-goanl number.
8. It can be partitioned 43 times with each term no longer than 2 and 631 times with each term no larger than 3.
9. It can be expressed as the sum of first 3 powers of 4 uniquely.
84 = 4^1 + 4^2 + 4^3
10. It is the maximal number of regions into which space can be divided by 7 spheres.
11. The probability of a pregnant woman having twins is 1 : 84.
12. According to Greek Anthalogy , Diophantus the Greek mathematician lived for 84 years. There is no other information known about his life except that comes from this problem which gives the result 84 on solving. The problem says---
This tomb holds Diophantus. Ah, how great a marvel! The tomb tells scientifically the measure of his life. God granted him to be a boy for 1/6 th of his life; and adding a twelfth part to this, he clothed his cheeks with down. He lit him the light of wedlock after a seventh part, and five years after his marriage he gave him a son. Alas, late born wretched child! After attaining the measure of half his father’s life, chill Fate took him. After consoling his grief by the study of numbers for four years, Diophantus ended his life. If Diophantus age is x years then we can write this algebraically as-- Which when solved gives x = 84. Diophanuts is better known as the father of algebra.
Mathematically 83
Eighty Three (83)
1. It is a Prime number which can be expressed as the sum of 3 consecutive primes as well as the sum of 5 consecutive primes.
83 = 23 + 29 + 31
= 11 + 13 + 17 + 19 + 23
2. It is also the sum of first 3 primes that ends with 1.
83 = 11 + 31 + 41
3. The atomic number of Bismuth is 83. It is the heaviest stable elements.
4. It can be portioned 42 times with each term not larger than 2.
5. It is a Sophie Germain Prime number. A prime number p is a Sophie Germain prime if 2p +1 is also a prime number. For p = 83 , 2p + 1 = 167 is also a prime number. It is named after the French mathematician Marie Sophie Germain.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a Prime number which can be expressed as the sum of 3 consecutive primes as well as the sum of 5 consecutive primes.
83 = 23 + 29 + 31
= 11 + 13 + 17 + 19 + 23
2. It is also the sum of first 3 primes that ends with 1.
83 = 11 + 31 + 41
3. The atomic number of Bismuth is 83. It is the heaviest stable elements.
4. It can be portioned 42 times with each term not larger than 2.
5. It is a Sophie Germain Prime number. A prime number p is a Sophie Germain prime if 2p +1 is also a prime number. For p = 83 , 2p + 1 = 167 is also a prime number. It is named after the French mathematician Marie Sophie Germain.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 82
Eighty two (82)
1. It is the product of two primes with the unique property that the sum of digits of 82 is equal to the product of sum of digits of its prime factors.
82 = 2 x 41 and 8+ 2 = 10 = 2 x ( 4 + 1)
2. It can be expressed as the sum of two primes in 5 different ways.
82 = 3 + 79
= 11 + 71
= 23 + 59
= 29 + 53
= 41 + 41
3. It is a 15 gonal number as well as a centered 27- gonal number.
4. Six hexagons can be linked in 82 different ways. A figure made from 6 hexagons is called hexahex. One arrangement is shown here.
5. The atomic number of Lead is 82.
6. On Ternary base it is a Palindrome digits. (10001)3 = (82)10.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the product of two primes with the unique property that the sum of digits of 82 is equal to the product of sum of digits of its prime factors.
82 = 2 x 41 and 8+ 2 = 10 = 2 x ( 4 + 1)
2. It can be expressed as the sum of two primes in 5 different ways.
82 = 3 + 79
= 11 + 71
= 23 + 59
= 29 + 53
= 41 + 41
3. It is a 15 gonal number as well as a centered 27- gonal number.
4. Six hexagons can be linked in 82 different ways. A figure made from 6 hexagons is called hexahex. One arrangement is shown here.
5. The atomic number of Lead is 82.
6. On Ternary base it is a Palindrome digits. (10001)3 = (82)10.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Matheamatically 81
Eighty One (81)
1. It is a square number.
9^2 = 81
2. It is the sum of first nine odd integers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
3. It is the square of sum of its digits. Apart from 0 and 1 it is the only number with this property.
81 = ( 8 + 1)^2
4. It is the fourth power of a number.
3^4 = 81
5. It is the smallest square whose sum of divisors is also a square number. The divisors of 81 are 1, 3, 9, 27 and 81 and their sum is 121 which is a square number.
1 + 3 + 9 + 27 + 81= 121 = 11^2
6. It is the sum of 9 consecutive numbers.
5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 = 81
7. The fraction 1/ 81= 0.01245679 012345679… is a repeating decimal containing every digits from 0 to 9 except 8.
8. On base 10, it is a Harshad number.
9. It is a heptagonal number as well as centered octagonal number.
10. The atomic number of Thallium is 81.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a square number.
9^2 = 81
2. It is the sum of first nine odd integers.
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81
3. It is the square of sum of its digits. Apart from 0 and 1 it is the only number with this property.
81 = ( 8 + 1)^2
4. It is the fourth power of a number.
3^4 = 81
5. It is the smallest square whose sum of divisors is also a square number. The divisors of 81 are 1, 3, 9, 27 and 81 and their sum is 121 which is a square number.
1 + 3 + 9 + 27 + 81= 121 = 11^2
6. It is the sum of 9 consecutive numbers.
5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 = 81
7. The fraction 1/ 81= 0.01245679 012345679… is a repeating decimal containing every digits from 0 to 9 except 8.
8. On base 10, it is a Harshad number.
9. It is a heptagonal number as well as centered octagonal number.
10. The atomic number of Thallium is 81.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 80
Eighty (80)
1. It is the product of 5 primes.
80 = 2 x 2 x 2 x 2 x 5
2. It is a semi perfect number. There are several divisors of 80 –1, 2, 4, 5, 8, 10, 16, 20, 40 and 80.
Adding some of the divisors we get 80 making it a semi perfect number.
80 = 1 + 4 + 5 + 10 + 20 + 40
3. It is the smallest number with the property that n and n+1 are both product of 4 or more primes.
80 = 2 x 2 x 2 x 2 x 5
81 = 3 x 3 x 3 x 3
4. It can be expressed as the sum of two primes in 4 distinct ways.
80 = 7 + 73 = 13 + 67 = 19 + 61 = 37 + 43
5. It is an abundant number.
6. It is a Harshad number.
7. The atomic number of Mercury is 80.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the product of 5 primes.
80 = 2 x 2 x 2 x 2 x 5
2. It is a semi perfect number. There are several divisors of 80 –1, 2, 4, 5, 8, 10, 16, 20, 40 and 80.
Adding some of the divisors we get 80 making it a semi perfect number.
80 = 1 + 4 + 5 + 10 + 20 + 40
3. It is the smallest number with the property that n and n+1 are both product of 4 or more primes.
80 = 2 x 2 x 2 x 2 x 5
81 = 3 x 3 x 3 x 3
4. It can be expressed as the sum of two primes in 4 distinct ways.
80 = 7 + 73 = 13 + 67 = 19 + 61 = 37 + 43
5. It is an abundant number.
6. It is a Harshad number.
7. The atomic number of Mercury is 80.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Matheamatically 79
Seventy Nine (79)
1. It is an odd prime number.
2. It is the sum of three consecutive prime. More interestingly, its reverse 97 is the sum of the reversible prime.
79 = 11 + 31 + 37
97 = 11 + 13 + 73
3. It is a Gaussian prime that can be written in the form of 4n + 3.
79 = 4 x 19 + 3
4. It can be written as the sum of their digits added to the product of their digits.
79 = (7 + 9) + (7 x 9)
In fact, all primes ending with 9 can be expressed as the sum of their digits added to the product of their digits.
5. It is a Pillai prime because 23! + 1 is divisible by 79, but 79 is not one more than a multiple of 23.
6. The atomic number of Gold is 79.
7. It is the maximal number of regions into which 12 lines divide a plane.
8. It is a 13-gonal as well as 26-gonal number.
9. It is a Kynea number. A Kynea number is an integer written in the form of .
The Kenya numbers are 7, 23, 79, 287, 1087...
10. It is a sexy prime number. Sexy primes are prime numbers that differ by 6.
The first 5 sexy prime pairs are – (5, 11), (7, 13), (11, 17), (13, 19), (17, 23) ---
There are infinite many sexy prime triplets and quadruplets numbers. The sexy prime triplets are three prime numbers that differ by 6 and the same happens with the quadruplets. (7, 13, 19), (17, 23, 29), (31, 37, 43) are the examples of sexy prime triplets whereas (5, 11, 17, 23), (11, 17, 23, 29)... are the example of sexy prime quadruplets.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is an odd prime number.
2. It is the sum of three consecutive prime. More interestingly, its reverse 97 is the sum of the reversible prime.
79 = 11 + 31 + 37
97 = 11 + 13 + 73
3. It is a Gaussian prime that can be written in the form of 4n + 3.
79 = 4 x 19 + 3
4. It can be written as the sum of their digits added to the product of their digits.
79 = (7 + 9) + (7 x 9)
In fact, all primes ending with 9 can be expressed as the sum of their digits added to the product of their digits.
5. It is a Pillai prime because 23! + 1 is divisible by 79, but 79 is not one more than a multiple of 23.
6. The atomic number of Gold is 79.
7. It is the maximal number of regions into which 12 lines divide a plane.
8. It is a 13-gonal as well as 26-gonal number.
9. It is a Kynea number. A Kynea number is an integer written in the form of .
The Kenya numbers are 7, 23, 79, 287, 1087...
10. It is a sexy prime number. Sexy primes are prime numbers that differ by 6.
The first 5 sexy prime pairs are – (5, 11), (7, 13), (11, 17), (13, 19), (17, 23) ---
There are infinite many sexy prime triplets and quadruplets numbers. The sexy prime triplets are three prime numbers that differ by 6 and the same happens with the quadruplets. (7, 13, 19), (17, 23, 29), (31, 37, 43) are the examples of sexy prime triplets whereas (5, 11, 17, 23), (11, 17, 23, 29)... are the example of sexy prime quadruplets.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Sunday, February 7, 2016
Mathematically 78
Seventy Eight (78)
1. It is the smallest number which can be written as the sum of four distinct squares in 3 ways.
78 = 1^2 + 2^2 +3^2 + 8^2
= 1^2 + 4^2 + 5^2 + 6^2
= 2^2 + 3^2 + 4^2 + 7^2
2. It is a triangular number.
3. It is the sum of first 12 natural numbers.
1 + 2 + 3 + --- + 12 = 78
4. There are 78 Tarot cards which is used for fortune telling. 56 among them the cards of lesser arcana and 22 are of greater arcana.
5. It can be written as the sum of two consecutive primes.
78 = 37 + 41
6. It is a 27-gonal number.
7. It is divisible by three distinct prime number 2, 3 and 13 which makes it a member of Sphenic number. A Sphenic number is a positive integer having three distinct prime factors.
The first five Sphenic numbers are – 30, 42, 66, 70 and 78.
8. The atomic number of Platinum is 78.
9. The number of chromosomes in canine of DNA is 78.
10. It is an abundant number.
Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the smallest number which can be written as the sum of four distinct squares in 3 ways.
78 = 1^2 + 2^2 +3^2 + 8^2
= 1^2 + 4^2 + 5^2 + 6^2
= 2^2 + 3^2 + 4^2 + 7^2
2. It is a triangular number.
3. It is the sum of first 12 natural numbers.
1 + 2 + 3 + --- + 12 = 78
4. There are 78 Tarot cards which is used for fortune telling. 56 among them the cards of lesser arcana and 22 are of greater arcana.
5. It can be written as the sum of two consecutive primes.
78 = 37 + 41
6. It is a 27-gonal number.
7. It is divisible by three distinct prime number 2, 3 and 13 which makes it a member of Sphenic number. A Sphenic number is a positive integer having three distinct prime factors.
The first five Sphenic numbers are – 30, 42, 66, 70 and 78.
8. The atomic number of Platinum is 78.
9. The number of chromosomes in canine of DNA is 78.
10. It is an abundant number.
Rajesh Kumar Thakur
rkthakur1974@gmail.com
Matheamatically 77
Seventy Seven (77)
1. It is the product of two odd primes.
77 = 7 x 11
2. It is the sum of first 8 prime numbers.
77 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19
3. Any number that is based on the pattern abcabc will be divisible by 77.
123123 ÷ 77 = 1599
4. It is the largest number that cannot be written as the sum of distinct numbers whose reciprocals add up to 1.
78 = 2 + 6 + 8 + 10 + 12 + 40
1 = ½ + 1/6 + 1/8 + 1/10 + 1/12 + 1/40
5. FORTRAN 77 was a computer programming developed by IBM in 1950.
6. It can be expressed as the sum of squares of three consecutive numbers.
77 = 4^2 + 5^2 + 5^2
7. The atomic number of Iridium is 77.
8. It can be partitioned 39 times with each tern not larger than 2 and 533 times with each term no larger than 2.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the product of two odd primes.
77 = 7 x 11
2. It is the sum of first 8 prime numbers.
77 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19
3. Any number that is based on the pattern abcabc will be divisible by 77.
123123 ÷ 77 = 1599
4. It is the largest number that cannot be written as the sum of distinct numbers whose reciprocals add up to 1.
78 = 2 + 6 + 8 + 10 + 12 + 40
1 = ½ + 1/6 + 1/8 + 1/10 + 1/12 + 1/40
5. FORTRAN 77 was a computer programming developed by IBM in 1950.
6. It can be expressed as the sum of squares of three consecutive numbers.
77 = 4^2 + 5^2 + 5^2
7. The atomic number of Iridium is 77.
8. It can be partitioned 39 times with each tern not larger than 2 and 533 times with each term no larger than 2.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 76
Seventy Six (76)
1. It is a Lucas number. The sequence in Lucas also moves like Fibonacci sequence. The numbers in Lucas series are- 1, 3, 4, 7, 11, 18, 29, 47, 76..
2. Sum of 4 consecutive Fibonacci sequence is 76.
8 + 13 + 21 + 34 = 76
3. It can be expressed as the sum of squares of primes.
76 = 32 + 32 + 32 + 72
4. Any power of 76 end with 76. Such numbers whose any power ends with the number itself is called automorphic. There are only two automorphic numbers 25 and 76 that are below 100.
76^2 = 5776 76^3 = 438976
76^4 = 33362176 76^5 = 2535525376
5. The atomic number of Osmium is 76.
6. It is a 14-gonal number.
7. It is a centered 25-gonal number.
8. It is a centered 25-gonal number.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is a Lucas number. The sequence in Lucas also moves like Fibonacci sequence. The numbers in Lucas series are- 1, 3, 4, 7, 11, 18, 29, 47, 76..
2. Sum of 4 consecutive Fibonacci sequence is 76.
8 + 13 + 21 + 34 = 76
3. It can be expressed as the sum of squares of primes.
76 = 32 + 32 + 32 + 72
4. Any power of 76 end with 76. Such numbers whose any power ends with the number itself is called automorphic. There are only two automorphic numbers 25 and 76 that are below 100.
76^2 = 5776 76^3 = 438976
76^4 = 33362176 76^5 = 2535525376
5. The atomic number of Osmium is 76.
6. It is a 14-gonal number.
7. It is a centered 25-gonal number.
8. It is a centered 25-gonal number.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Sunday, January 31, 2016
Mathematically 75
Seventy Five (75)
1. It is also called three quarter. 3/ 4 = 75%
2. It is one less than the sum of squares of its digit.
75 = 7^2 + 5^2 – 1
3. It can be expressed as the sum of 10 consecutive primes beginning from 3.
75 = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
4. It is a nonagonal number.
5. It is a pentagonal pyramid number. A pentagonal pyrammdal number is a figurarive number that represents the number of objects in a pyramid with pentagonal base.
75 = 1 + 6 + 18 + 40
6. The atomic number of Rbenium is 75.
7. It is a Keith number. A Keith numbers run like a Fibonacci sequence. 75 has its digit 7 and 5. If we start to form a sequence with the digit 75 and go on finding the next number by adding the previous two as done in Fibonacci sequence we get 75.
7, 5 , 12, 17, 29, 46, 75.. It is named after Mike Keith who discovered this number in 1987
8. It is a Self-number. A self-number is an integer which in a given base cannot be generated by any other integer added to the sum of that other integer’s digit.
Example:- 21 is not a self-number as it can be generated by 15 and sum of tis digit 1 and 5, whereas 20 is a self-number as it cannot be generated by any such numbers.
9. A diamond anniversary is celebrated after every 75 years.
10. The Proper divisor of 75 are - 1, 3, 5, 15, 25 whose sum is a perfect square
1 + 3 + 5 + 15 + 25 = 49 = 7^2
Moreover , its product
1 x 3 x 5 x 15 x 25 = 5625 = 75^2
is also a square.
11. 2^75 + 75 is a prime
Dr Rajesh Kumar Thakur
1. It is also called three quarter. 3/ 4 = 75%
2. It is one less than the sum of squares of its digit.
75 = 7^2 + 5^2 – 1
3. It can be expressed as the sum of 10 consecutive primes beginning from 3.
75 = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
4. It is a nonagonal number.
5. It is a pentagonal pyramid number. A pentagonal pyrammdal number is a figurarive number that represents the number of objects in a pyramid with pentagonal base.
75 = 1 + 6 + 18 + 40
6. The atomic number of Rbenium is 75.
7. It is a Keith number. A Keith numbers run like a Fibonacci sequence. 75 has its digit 7 and 5. If we start to form a sequence with the digit 75 and go on finding the next number by adding the previous two as done in Fibonacci sequence we get 75.
7, 5 , 12, 17, 29, 46, 75.. It is named after Mike Keith who discovered this number in 1987
8. It is a Self-number. A self-number is an integer which in a given base cannot be generated by any other integer added to the sum of that other integer’s digit.
Example:- 21 is not a self-number as it can be generated by 15 and sum of tis digit 1 and 5, whereas 20 is a self-number as it cannot be generated by any such numbers.
9. A diamond anniversary is celebrated after every 75 years.
10. The Proper divisor of 75 are - 1, 3, 5, 15, 25 whose sum is a perfect square
1 + 3 + 5 + 15 + 25 = 49 = 7^2
Moreover , its product
1 x 3 x 5 x 15 x 25 = 5625 = 75^2
is also a square.
11. 2^75 + 75 is a prime
Dr Rajesh Kumar Thakur
Matheamatically 74
Seventy Four (74)
1. The maximal number of regions into which a plane can be divided by 9 circles are 74.
2. It is the product of two distinct prime.
74 = 2 x 37
3. It is the sum of four consecutive numbers.
74 = 17 + 18 + 19 + 20
4. The atomic number of Tungsten is 74.
Regards
Dr Rajesh Kumar Thakur
1. The maximal number of regions into which a plane can be divided by 9 circles are 74.
2. It is the product of two distinct prime.
74 = 2 x 37
3. It is the sum of four consecutive numbers.
74 = 17 + 18 + 19 + 20
4. The atomic number of Tungsten is 74.
Regards
Dr Rajesh Kumar Thakur
Mathematically 73
Seventy Three (73)
1. It is the 21st prime number. It along with the next prime 73 makes a twin prime.
2. It is the smallest number besides 1 which is 1 less than the double of its reverse.
73 = 2 x 37 – 1
3. It is the smallest number with the property that its neighbours can be written as the sum of two squares.
72 = 6^2 + 6^2
73 = 3^2 + 8^2
74 = 5^2 + 8^2
4. It is the smallest prime congruent to 1 modulo 24.
73 ≡ 1 (mod 24)
5. On base 8 it is the only number with repeated unit.
73 = ( 111 ) 8
6. The square root of 73 is 8. 544.. which when summed up gives 21 that is the product of its digit.
(73)^1/2 = 8. 544 and 8 + 5 + 4 + 4 = 7 x 3
7. It is a centered 12 gonal and 24 gonal number.
8. It is the least number of sixth powers needed to represent every possible integer.
9. The atomic number of Tantalum is 73.
10. There is a very interesting puzzle whose final result comes out to be 73.
The steps of the puzzles are--
-a) Take any four digit number
b) Write this number again making a eight digit number
c) Divide this 8 digit number by 137
d) divide the quotient by the 4 digit number taken initially.
The final result will be 73.
Example: - Let the four digit number taken is 1238.
Make this number an eight digit number by writing this number twice, i.e. 12381238.
Divide this big number by 137 and you get the quotient 90374 which when divided by 1238 gives you 73.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is the 21st prime number. It along with the next prime 73 makes a twin prime.
2. It is the smallest number besides 1 which is 1 less than the double of its reverse.
73 = 2 x 37 – 1
3. It is the smallest number with the property that its neighbours can be written as the sum of two squares.
72 = 6^2 + 6^2
73 = 3^2 + 8^2
74 = 5^2 + 8^2
4. It is the smallest prime congruent to 1 modulo 24.
73 ≡ 1 (mod 24)
5. On base 8 it is the only number with repeated unit.
73 = ( 111 ) 8
6. The square root of 73 is 8. 544.. which when summed up gives 21 that is the product of its digit.
(73)^1/2 = 8. 544 and 8 + 5 + 4 + 4 = 7 x 3
7. It is a centered 12 gonal and 24 gonal number.
8. It is the least number of sixth powers needed to represent every possible integer.
9. The atomic number of Tantalum is 73.
10. There is a very interesting puzzle whose final result comes out to be 73.
The steps of the puzzles are--
-a) Take any four digit number
b) Write this number again making a eight digit number
c) Divide this 8 digit number by 137
d) divide the quotient by the 4 digit number taken initially.
The final result will be 73.
Example: - Let the four digit number taken is 1238.
Make this number an eight digit number by writing this number twice, i.e. 12381238.
Divide this big number by 137 and you get the quotient 90374 which when divided by 1238 gives you 73.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 72
Seventy Two (72)
1. It is equal to half of the gross.
½ x 144 = 72
2. The sum of first eight even number is 72.
2 + 4 + 6 + 8 +10 + 12 + 14 + 16 = 72
3. It is the only number which is eight times the sum of its own digits.
72 = 8 x (7 + 2)
4. It is the smallest number that can be written as the sum of two prime number in two distinct ways and the most interesting part of this partition is that the primes used ends with 1.
72 = 11 + 61
= 31 + 41
5. The normal human pulse rate is 72 beats/ minutes at rest.
6. An isosceles triangle having a pair of 72 degree angle is called Golden triangle.
7. It is one- fifth of a central angle (360 degree) of circle.
8. The internal angle of a regular pentagon is 72 degree.
9. It is divisible by every number from 1 to 10 apart from 5, 7 and 10.
10. It is the smallest 5th power equal to the sum of 5 other 5th powers.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
11. In financial circle rule of 72 is a simple method to quickly estimate compound interest. If you want to know how long will it take to double your money, divide 72 by the interest rate and the answer will be the number of years.
12. It is the sum of four consecutive primes as well as the sum of six consecutive prime number.
72 = 13 + 17 + 19 + 23 (sum of 4 consecutive numbers)
= 5 + 7 + 11 + 13 + 17 + 19 (sum of 6 consecutive numbers)
13. It is a Harshad number on base 10.
14. The atomic number of Hafnium is 72.
15. It is an Abundant number because the sum of factors of 72 excluding itself is more than 72. The factors are- 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
16. It is a 25- gonal number.
17. It is the maximum number of sphere that can touch another sphere in a lattice packing in 6 dimensions.
18. The great Rhombicuboctahedron in an Archimedean solid is 72 edges.
19. The sum of the eight row of Lozanic triangle is 72.
72 = 1 + 4 + 12 + 19 + 19 + 12 + 4 + 1
The Lozanic triangle is a triangular array of binomial coefficients similar to Pascal’s triangle. It is named after Sima Lozanic, the Serbian chemist.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
1. It is equal to half of the gross.
½ x 144 = 72
2. The sum of first eight even number is 72.
2 + 4 + 6 + 8 +10 + 12 + 14 + 16 = 72
3. It is the only number which is eight times the sum of its own digits.
72 = 8 x (7 + 2)
4. It is the smallest number that can be written as the sum of two prime number in two distinct ways and the most interesting part of this partition is that the primes used ends with 1.
72 = 11 + 61
= 31 + 41
5. The normal human pulse rate is 72 beats/ minutes at rest.
6. An isosceles triangle having a pair of 72 degree angle is called Golden triangle.
7. It is one- fifth of a central angle (360 degree) of circle.
8. The internal angle of a regular pentagon is 72 degree.
9. It is divisible by every number from 1 to 10 apart from 5, 7 and 10.
10. It is the smallest 5th power equal to the sum of 5 other 5th powers.
72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
11. In financial circle rule of 72 is a simple method to quickly estimate compound interest. If you want to know how long will it take to double your money, divide 72 by the interest rate and the answer will be the number of years.
12. It is the sum of four consecutive primes as well as the sum of six consecutive prime number.
72 = 13 + 17 + 19 + 23 (sum of 4 consecutive numbers)
= 5 + 7 + 11 + 13 + 17 + 19 (sum of 6 consecutive numbers)
13. It is a Harshad number on base 10.
14. The atomic number of Hafnium is 72.
15. It is an Abundant number because the sum of factors of 72 excluding itself is more than 72. The factors are- 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
16. It is a 25- gonal number.
17. It is the maximum number of sphere that can touch another sphere in a lattice packing in 6 dimensions.
18. The great Rhombicuboctahedron in an Archimedean solid is 72 edges.
19. The sum of the eight row of Lozanic triangle is 72.
72 = 1 + 4 + 12 + 19 + 19 + 12 + 4 + 1
The Lozanic triangle is a triangular array of binomial coefficients similar to Pascal’s triangle. It is named after Sima Lozanic, the Serbian chemist.
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Mathematically 71
Seventy one (71)
1. It is a 20th Prime number which makes a twin prime with the next prime 73.
2. It is a Chen Prime. A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. It is named after Chen Jingrun. There are infinitely Chen primes and the first ten are- 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
3. It is also a Pillai prime named after the Indian mathematician Subbayya Sivasankaranarayana Pillai. A Pillai prime is a prime number p for which there is no integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. The first 10 Pillai primes are 23, 29, 59, 61, 67, 71, 79, 83, 109 and 137. Mathematically
4. It is a 5th centered heptagonal number. A centered heptagonal number is a centered figurative number that represents a heptagon with a dot in the centre and other dot surrounding in heptagonal layers. It is genereated by using the formula
5. The cube of 71 contains all odd numbers from 3 to 11.
71^3 = 357911
6. The cube of 71 is the only two digit cube that ends with 11
7. The atomic number of Lanthanide is 71.
8. It is one of the number in the last pair of Brocard number. Pairs of numbers (n ,m) that solves the Brocard’s problem which states to find the integral value of n for which this was posed by Henri Brocard in 1876 and later solved by Srinivasa Ramanujan independently in 1913 without knowing that this problem had been solved earlier. There are only three pairs of such number- (4, 5), (5, 11) and (7, 11).
9. Sum of all previous prime numbers is divisible by 71.
2 + 3 + 5 + 7 + 11+ --- + 67 is divisible by 71.
10. The square of 71 can be expressed the sum of digits of its factorial.
71^2 = 7! + 1!
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
1. It is a 20th Prime number which makes a twin prime with the next prime 73.
2. It is a Chen Prime. A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. It is named after Chen Jingrun. There are infinitely Chen primes and the first ten are- 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
3. It is also a Pillai prime named after the Indian mathematician Subbayya Sivasankaranarayana Pillai. A Pillai prime is a prime number p for which there is no integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. The first 10 Pillai primes are 23, 29, 59, 61, 67, 71, 79, 83, 109 and 137. Mathematically
4. It is a 5th centered heptagonal number. A centered heptagonal number is a centered figurative number that represents a heptagon with a dot in the centre and other dot surrounding in heptagonal layers. It is genereated by using the formula
5. The cube of 71 contains all odd numbers from 3 to 11.
71^3 = 357911
6. The cube of 71 is the only two digit cube that ends with 11
7. The atomic number of Lanthanide is 71.
8. It is one of the number in the last pair of Brocard number. Pairs of numbers (n ,m) that solves the Brocard’s problem which states to find the integral value of n for which this was posed by Henri Brocard in 1876 and later solved by Srinivasa Ramanujan independently in 1913 without knowing that this problem had been solved earlier. There are only three pairs of such number- (4, 5), (5, 11) and (7, 11).
9. Sum of all previous prime numbers is divisible by 71.
2 + 3 + 5 + 7 + 11+ --- + 67 is divisible by 71.
10. The square of 71 can be expressed the sum of digits of its factorial.
71^2 = 7! + 1!
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
Mathematically 70
Seventy (70)
1. The sum of divisors of 70 including itself is a square number.
70 has the factors- 1, 2, 5, 7, 10,14, 35 and 70 which when added gives 144 which is a square number. 1 + 2 + 5 + 7 + 10 + 14 + 35 + 70 = 144 = 12^2
2. It is the smallest weird number. A number is called weird if it is an abundant number without being the sum of any set of its divisors. As discussed above that 1, 2, 5, 7, 10, 14 , 35 and 70 are the factors of 70 whose sum excluding 70 is 74 but no set of these factors sum to 70.
1 + 2 + 5 + 7 + 10 + 14 + 35 = 74
35 + 14 + 10 + 7 + 5 = 71
35 + 14 + 10 + 7 + 5 = 63
There are very few Weird number and you will be astonished to know that there are only 7 weird number below 10000 and they are 70, 836, 4030, 5830, 7192, 7912 and 9272.
3. The sum of squares of first 24 natural numbers is the square of 70.
1^2 + 2^2 + 3^2 + … + 24^2 = 70^2
4. It is a Pell number. Pell numbers are similar to Fibonacci numbers and are generated by the formula
An = 2 An – 1 + An – 2
The first 10 Pell numbers are 1, 2, 5, 12, 29, 70, 169, 408, 985 and 2378.
5. In base 10 it is a Harshad number as it is divisible by sum of its digit.
6. It is the seventh pentagonal number.
7. The atomic number of Yttterbium is 70.
8. 270 = 1180591620717411303424. The sum of the digits is also 70,
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
1. The sum of divisors of 70 including itself is a square number.
70 has the factors- 1, 2, 5, 7, 10,14, 35 and 70 which when added gives 144 which is a square number. 1 + 2 + 5 + 7 + 10 + 14 + 35 + 70 = 144 = 12^2
2. It is the smallest weird number. A number is called weird if it is an abundant number without being the sum of any set of its divisors. As discussed above that 1, 2, 5, 7, 10, 14 , 35 and 70 are the factors of 70 whose sum excluding 70 is 74 but no set of these factors sum to 70.
1 + 2 + 5 + 7 + 10 + 14 + 35 = 74
35 + 14 + 10 + 7 + 5 = 71
35 + 14 + 10 + 7 + 5 = 63
There are very few Weird number and you will be astonished to know that there are only 7 weird number below 10000 and they are 70, 836, 4030, 5830, 7192, 7912 and 9272.
3. The sum of squares of first 24 natural numbers is the square of 70.
1^2 + 2^2 + 3^2 + … + 24^2 = 70^2
4. It is a Pell number. Pell numbers are similar to Fibonacci numbers and are generated by the formula
An = 2 An – 1 + An – 2
The first 10 Pell numbers are 1, 2, 5, 12, 29, 70, 169, 408, 985 and 2378.
5. In base 10 it is a Harshad number as it is divisible by sum of its digit.
6. It is the seventh pentagonal number.
7. The atomic number of Yttterbium is 70.
8. 270 = 1180591620717411303424. The sum of the digits is also 70,
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
Wednesday, January 27, 2016
Mathematically 69
Sixty Nine (69)
1. It is the fifth number that stays same when written upside down.
2. It is the only number whose square and cube together use all the digits from 0 to 9.
69^2 = 4761
69^3 = 328509
3. 6 and 9 when rotated with 180 degree gives the same number. Moreover, the binary value of 6 and 9 are complement to each other.
(6)10 = 110 = (0110)2
(9)10 = (1001)2.
In binary number system 0 and 1 are complementary to each other here the complement of 0110 is 1001 which is nothing but the number 6 and 9 written in decimal system.
4. 10^69 + 69 and 10^69 - 69 is also prime number.
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
1. It is the fifth number that stays same when written upside down.
2. It is the only number whose square and cube together use all the digits from 0 to 9.
69^2 = 4761
69^3 = 328509
3. 6 and 9 when rotated with 180 degree gives the same number. Moreover, the binary value of 6 and 9 are complement to each other.
(6)10 = 110 = (0110)2
(9)10 = (1001)2.
In binary number system 0 and 1 are complementary to each other here the complement of 0110 is 1001 which is nothing but the number 6 and 9 written in decimal system.
4. 10^69 + 69 and 10^69 - 69 is also prime number.
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
Matheamatically 68
Sixty Eight (68)
1. It is the smallest composite number that becomes prime by turning it upside down. 68 turning upside down gives 89 which in a prime number.
2. It is the sum of 8 consecutive numbers beginning from 5.
68 = 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
3. It is a Truncated Tetrahedral number.
4. There are exactly 68 ten digit binary numbers in which each digit is the same as one of it's adjacent digits.
5. The atomic number of Erbium is 68.
6. 68 is the largest known number to be the sum of two primes in exactly two different ways:
68 =7+61
=31+37.
7. In a normal distribution around 68% of value are within one standard deviation away from the mean.
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
1. It is the smallest composite number that becomes prime by turning it upside down. 68 turning upside down gives 89 which in a prime number.
2. It is the sum of 8 consecutive numbers beginning from 5.
68 = 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12
3. It is a Truncated Tetrahedral number.
4. There are exactly 68 ten digit binary numbers in which each digit is the same as one of it's adjacent digits.
5. The atomic number of Erbium is 68.
6. 68 is the largest known number to be the sum of two primes in exactly two different ways:
68 =7+61
=31+37.
7. In a normal distribution around 68% of value are within one standard deviation away from the mean.
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
Mathematically 67
Sixty Seven (67)
1. It is the smallest number which is both palindromic in base 5 and 6.
(67)10 = (232)5 = (151)6
2. The sum of 5 consecutive prime numbers starting from 7 is 67.
7 + 11 + 13 + 17 + 19 = 67
3. The cousin prime of 67 is 71. Two primes that differ by 4 is called Cousin Primes.
(3, 7), (7, 11), (13, 17)... etc. are cousin primes
4. The atomic number of Homium is 67.
5. It is the maximal number of regions into which 11 lines divides a plane.
6. It is a centered 11- gonal and 22- gonal number.
7. A prime number with a 7 in it form an interesting pattern.
67 x 67 = 4489
667 x 667 = 444889
6667 x 6667 = 44448889
8. More interesting fact
-----------------------------------------------------
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
1. It is the smallest number which is both palindromic in base 5 and 6.
(67)10 = (232)5 = (151)6
2. The sum of 5 consecutive prime numbers starting from 7 is 67.
7 + 11 + 13 + 17 + 19 = 67
3. The cousin prime of 67 is 71. Two primes that differ by 4 is called Cousin Primes.
(3, 7), (7, 11), (13, 17)... etc. are cousin primes
4. The atomic number of Homium is 67.
5. It is the maximal number of regions into which 11 lines divides a plane.
6. It is a centered 11- gonal and 22- gonal number.
7. A prime number with a 7 in it form an interesting pattern.
67 x 67 = 4489
667 x 667 = 444889
6667 x 6667 = 44448889
8. More interesting fact
-----------------------------------------------------
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
Mathematically 66
Sixty Six ( 66 )
• It is the sum of first 11 natural numbers.
1 + 2 +….+ 11 = 66
• It is a triangular number.
• There are 66 blocks in the given skeleton tower. This tower is six blocks high and has been built with 66 blocks.
(Courtesy :- Richard Philips :- Number)
• It is a Palindrome number.
• It is a Hexagonal number.
• It can be written as the product of three primes.
66 = 2 x 3 x 11
• The atomic number of Dysprosium is 66.
• 66 has seven divisors including itself. If you sum up all these divisors then you get a square number.
1 + 2 + 3 + 6 + 11 + 22 + 33 + 66 = 144 = 12^2
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
• It is the sum of first 11 natural numbers.
1 + 2 +….+ 11 = 66
• It is a triangular number.
• There are 66 blocks in the given skeleton tower. This tower is six blocks high and has been built with 66 blocks.
• It is a Palindrome number.
• It is a Hexagonal number.
• It can be written as the product of three primes.
66 = 2 x 3 x 11
• The atomic number of Dysprosium is 66.
• 66 has seven divisors including itself. If you sum up all these divisors then you get a square number.
1 + 2 + 3 + 6 + 11 + 22 + 33 + 66 = 144 = 12^2
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com
Monday, January 25, 2016
Mathematically 65
Sixty Five (65)
1. It is the product of two odd primes.
65 = 5 x 13
2. It is the lowest integer that becomes a square if its digit reversed is either added or subtracted from it.
65 – 56 = 9 = 3 x 3
65 + 56 = 121 = 11 x 11
3. The difference of square number of 65 and its reverse 56 is also a square number and the trio forms a Pythagorean triplets.
65^2 – 56^2 = 33^2
4. In the ASCII code used in computer 65 represents A.
5. It is the fifth star number.
6. It is a Rhombic Dodecahedral number.
7. It is a octagonal number
8. The atomic number of Terbium is 65.
9. The numbers of Euler’s idoneal numbers are 65. In mathematics, Euler's idoneal numbers, are the positive integers D such that any integer expressible in only one way as x^2 ± Dy^2 (where x^2 is relatively prime to Dy^2) is a prime, prime power, or twice one of these. The 65 idoneal numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848
10. It is the length of the hypotenuse of 4 different Pythagorean triangles.
65^2 = 16^2 + 63^2
= 33^2 + 56^2
= 39^2 + 52^2
= 25^2 + 60^2
11. It can be expressed uniquely in power of 5 consecutive numbers with 5 consecutive powers in reversed order.
65=15+24+33+42+51
12. It is a magic constant of 5 x 5 magic square.
13. The sum of 10 natural numbers starting from 2 is 65.
2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =65
14. The sum of numbers in each ring of the concentric circle in the given magic circle is 65.
9 + 23 + 12 + 1 + 29 = 65
2 + 16 + 10 + 24 + 13 = 65
6 + 17 + 3 + 14 + 25 = 65
4 + 15 + 21 + 7 + 18 = 65
11 + 5 + 19 + 8 + 22 = 65
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
1. It is the product of two odd primes.
65 = 5 x 13
2. It is the lowest integer that becomes a square if its digit reversed is either added or subtracted from it.
65 – 56 = 9 = 3 x 3
65 + 56 = 121 = 11 x 11
3. The difference of square number of 65 and its reverse 56 is also a square number and the trio forms a Pythagorean triplets.
65^2 – 56^2 = 33^2
4. In the ASCII code used in computer 65 represents A.
5. It is the fifth star number.
6. It is a Rhombic Dodecahedral number.
7. It is a octagonal number
8. The atomic number of Terbium is 65.
9. The numbers of Euler’s idoneal numbers are 65. In mathematics, Euler's idoneal numbers, are the positive integers D such that any integer expressible in only one way as x^2 ± Dy^2 (where x^2 is relatively prime to Dy^2) is a prime, prime power, or twice one of these. The 65 idoneal numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848
10. It is the length of the hypotenuse of 4 different Pythagorean triangles.
65^2 = 16^2 + 63^2
= 33^2 + 56^2
= 39^2 + 52^2
= 25^2 + 60^2
11. It can be expressed uniquely in power of 5 consecutive numbers with 5 consecutive powers in reversed order.
65=15+24+33+42+51
12. It is a magic constant of 5 x 5 magic square.
13. The sum of 10 natural numbers starting from 2 is 65.
2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =65
14. The sum of numbers in each ring of the concentric circle in the given magic circle is 65.
9 + 23 + 12 + 1 + 29 = 65
2 + 16 + 10 + 24 + 13 = 65
6 + 17 + 3 + 14 + 25 = 65
4 + 15 + 21 + 7 + 18 = 65
11 + 5 + 19 + 8 + 22 = 65
DR RAJESH KUMAR THAKUR
rkthakur1974@gmail.com
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