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