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

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

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

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

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
67=(110)+(111)+(112)
      = 1 + 11 + 55

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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

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