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
Sunday, March 20, 2016
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
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