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
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Dr Rajesh Kumar Thakur
Saturday, February 27, 2016
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
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