Mathematically 31

                                                     Thirty One (31)

There are exactly 31 positive integers which cannot be written as the sum of two distinct squares; and one of them is the number 31. Let's explore some of the properties of 31

1. It is a prime number.

2. It is the sum of five powers of 2.

2^0 + 2^1 + 2^2 + 2^3 + 2^4 = 31

3. It is the third Mersenne prime.
2^5 – 1 = 31

4. January, March, May, July, August, October, December have 31 days.

5. Sum of first 31 odd primes is a perfect square.
3 + 5 + 7 + 11 + …. + 83 + 87 + 89 = 31^2

6. It is a Centered Pentagonal number.

7. It is a repdigit in base 5 and base 2.
(111)5 = (11111)2 = 31

8. The numbers 31, 331, 3331, 33331, 333331, 3333331, and 33333331 are all prime but 333333331 is not a prime.

9. It can be partioned into 6842 ways.

10. It is amongst the two known number that can be written in 2 ways as the sum of successive powers starting with 1.
31 = 1 + 5 + 5^2                                               31= 1 + 2 + 2^2 + 2^3 +2^4

11. It is the sum of first two primes raised to themselves.

31 = 2^2 +3 ^3

12. The digit sum of 31st Fibonacci number (1346269) is 31
1 + 3 + 4 + 6 + 2 + 6 + 9 = 31

13. The number obtained by raising the first four prime to themselves comes out to be a prime.
22+33+55+77=826699


14 . It is a Happy Prime. A happy prime is a number that is both happy and prime.
The first happy primes are 7, 13, 19, 23, 31, 79 etc.
A happy number is defined in the following way:-- Start with any number, replace the number
by the sum of the squares of its digits. Repeat the process until you get 1.

15. It is the minimal move required to solve the Tower of Hanoi puzzle with 5 disks.

16. 31 is also the smallest integer that can be written as the sum of four positive squares in two ways

31 = 1+1+4+25

31 = 4+9+9+9.

17. 31 is an evil math teacher number. The sequence of  the maximum number of regions obtained by joining n points around a circle by straight lines begins 2, 4, 8, 16... but for five points, it is 31.


Send Your Comments to
Dr Rajesh Thakur
rkthakur1974@gmail.com

Mathematically 30

Thirty (30)

1. It is the product of first three prime numbers.
30 = 2 x 3 x 5

2. It is the sum of first fours square number.
30 = 1^2 + 2^2 + 3^2 + 4^2

3. It is a Pyramidal number. A pyramidal number is a figurative number that represents a pyramid with a base and a given number of sides.

4. It is the largest number such that every smaller co prime to it is a prime.

5. Dodecahedron and Icosahedrons both have 30 edges.

Dodecahedron Icosahedron
6. A polygon with 30 sides is called a tricontagon.

7. It is a Harshad number because 30 is divisible by the sum of its digit 3 + 0 = 3.

8. The atomic number of zinc is 30.

9. It is the area of right angle triangle with side 5, 12 and 13. It also has the perimeter 30.

10. It is the 5th abundant number.

11. N^5 – N is divisible by 30.

12. It is a round number. A number (n) is said to be a round number if it has no prime factor greater than square root n.


Enjoy reading

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

importance of 29

Twenty Nine (29)

1. It is the tenth prime number.

2. There are 29 days in the month of February in a leap year.

3. It is the smallest prime of the form 7n + 1, where n is any integer.
7 x 1 + 1 = 8 is not a prime
7 x 2 + 1 = 15 is not a prime
7 x 3 + 1 = 22 is not a prime
7 x 4 + 1 = 29 is a prime.

4. It is the sum of square of three consecutive numbers.
29 = 2^2 + 3^2 + 4^2

5. Probability of being born on 29th February is 1/ 1461

6. Seven straight cut through a pizza can divide it into as many as 29 pieces.

7. It is the seventh Lucas number. The series in Lucas number goes like- 1, 3, 4, 7, 11, 18, 29, 47, ---. Like Fibonacci series the next number in Lucas series also is the sum of previous two.

8. Twenty Nine is the only number written with as many strokes as the numerical value.

2 + 4 + 4 + 3 + 2 + 3 + 3 + 1 + 3 + 4

9. It is the third number following 1 and 5 for which 2n^2 – 1 is a square.
2 x 29^2 – 1 = 41^2

10. Saturn requires 29 years to orbit the Sun.

11. The atomic number of Copper is 29.

12. It is a Markov number. A Markov number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation x^2 + y^2 + z^2 = 3xyz. This equation was studied by Andrey Markov.
29 appears in the solution of the above equation in following manners.
{2, 5, 29 }, {2, 29, 169}, {5, 29, 433} etc.

13.The 29th day of the year; 2²⁹ = 536870912 a nine-digit number with no digit repeated. 

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Importance of Number 28

Twenty Eight (28)
1. It is a triangular number.

2. It is the 2nd perfect number. A number is said to be perfect if it is equal to the sum of its perfect divisors except itself.

28 has the factors 1, 2, 4, 7 and 14 that adds up to 28.
All known perfect numbers are even numbers as they end with 6 or 8.
The first five perfect numbers are- 6, 28, 496, 8128 and 33550336.

3. There are 28 days in the month of February except in a leap year when it has 29 days.

4. There are 28 dominoes in a standard double six set.

5. It is a hexagonal number. A hexagonal number is a figurative number.

The formula for the nth hexagonal number is given by --

6. It is the sum of first five consecutive primes.
28 = 2 + 3 + 5 + 7 + 11

7. A 28 side polygon is called Icosikaioctagon.

8. The Gregorian calendar repeats in 28 years. The calendar of 2013 will work in 2041.

9. It can be portioned into 3718 ways.

10. It is the third positive integer with a prime factorization in the form of 2^2q, where q is an odd prime.
28 = 2 x 2 x 7 = 2^2 x 7

11. It is used as an approximation to the number of days in the lunar month.

12. The atomic mass of Silicon is 28.

13. The atomic number of Nickel is 28.

14. The revolution time of the surface of the sun on itself is 28 days.


Send your valuable comments to
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Importance of Number 27

Twenty seven (27)
1. It is the smallest two digit number in which the sum of digits is equal to the sum of prime factors.

27 = 3 x 3 x 3 and 2 + 7 = 3 + 3 + 3

2. It is the largest number which is equal to the sum of the digits of its cube.
27^3 = 19683 and 1 + 9 + 6 + 8 + 3 = 27

3. It is the magic constant of the multiple of 1/7.

1/7 = 0. 142857… 2/7 = 0.285714-- 3/7 = 0.428571-- 4/7 = 0.571428-- 5/6 = 0.714285-- 6/7 = 0.857142--

Digit 1 4 2 8 5 7 Digit Sum
1 1 4 2 8 5 7 27
2 2 8 5 7 1 4 27
3 4 2 8 5 7 1 27
4 5 7 1 4 2 8 27
5 7 1 4 2 8 5 27
6 8 5 7 1 4 2 27
Column sum 27 27 27 27 27 27

4. It is the third decagonal number. A decagonal number represents a decagon. The nth decagonal number is given by the formula--

5. It is a cubic number.
27 = 3 x 3 x 3

6. It is the only positive integer that is 3 times the sum of its digits.
27 = 3 x (2 + 7)

7. In Hindu astrology there are 27 nakshatra (stars) each controls a section of the zodiac signs.

The first few are – Ashwini, Bharani, Kritika –

8. If you add up all the digits from 2 to 7 it will give you 27. It is the second two digit number besides 15 with this property.
2 + 3 + 4 + 5 + 6 + 7 = 27

9. The atomic number of Cobalt is 27.

10. The atomic weight of Aluminum is 27.

11. It is the first odd perfect cube apart from 1.

12. It is the smallest number that can be expressed as the sum of 3 squares in 2 ways.
27 = 3^2 + 3^2 + 3^2 27 = 5^2 + 1^2 + 1^2

13. There are 27 cubes in a Rubik’s


14. It can be portioned in 3010 ways.

Send Your Comments to

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Importance of Number 26

Twenty Six (26)

1. There are 26 letters in English which can make 40,329 x 1022 different combination.

2. It is the only number between a square number and a cube number.
25 = 5^2 < 26 < 27 = 3^3 

 3. It is the smallest number which is not a palindrome number but its square is a palindrome. A palindrome number read from left to right and vice versa gives the same result. 26^2 = 676 this is a palindrome number. 

 4. It is the sum of three consecutive Fibonacci number. 26 = 5 + 8 + 13 

 5. It is the difference of the 2nd pair of amicable numbers. Two numbers are said to be amicable if the sum of factors of one number excluding itself gives the other number and vice versa. The first pair of amicable number is 220 and 284. The second pair of amicable number is 1184 and 1210. 1210 – 1184 = 26 

 6. A rhombicuboctahedron has 26 sides. It is an Archimedean solid with 8 triangular and 18 square faces.

7. In ternary system it has repeated digit. 26 written on base three is equal to 2223.

8. The atomic number of Iron is 26.

9. It is the third heptagonal pyramidal number. The heptagonal number for n can be calculated by adding up the heptagonal numbers for 1 to n, or

by using the formula ----n(n + 1)(5n - 2)/6.

The first 5 heptagonal pyramidal numbers are – 1, 8, 26, 60 and 115.

10. It is the sum of its cubes.
26^3 = 17576 and 1 + 7 + 5 + 7 + 6 = 26

11. Here is a Heptagonal puzzle which is quite difficult to solve. This puzzle is due to Henry Dudeney (1847 -1930). The puzzle goes like this—using the number 1 to 14 place a different number in each circle so that the three numbers along every side add up to 26. Henry himself has given this solution.

Send Your comments to
Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Importance of 25

Twenty Five (25)
1. It is a centered octagonal number.

2. It is a square number.


3. It is the smallest square number that can be written as a sum of 2 consecutive squares.
25 = 3^2 + 4^2

4. It is the sum of first 5 odd numbers. 1 + 3 + 5 + 7 + 9 = 25

5. It is the sum of two triangular numbers.

+ =
6. If you place 6 with 25 in two different ways you get two square numbers.
625 = 25^2 and 256 = 16^2

7. A number ending with 25, 50, 75 or 00 will be divisible by 25.
Example:- 450, 625, 875, 1900 are divisible by 25.

8. It is the difference of squares of two consecutive numbers.
25 = 132 – 122

9. It is the sum of squares of two consecutive numbers.
25 = 32 + 42

10. It can be portioned in 1958 ways.

11. All powers of 25 ends with 25

25^2 = 625
25^3 = 15625

12. It is the only solution of (n – 1 )! + 1 = n^k
25 = 5^2 = (5 – 1 )! + 1


Send your comments to
rkthakur1974@gmail.com

Dr Rajesh Kumar Thakur