Question

What percentage of the first 500 natural number can be written as the difference of two perfect square?

Answer 1

All The odd numbers can necessarily be written as difference of two squares. For example 1 = (1)^2 - (0)^2, 9 = (5)^2 - (4)^2 etc.

 

Besides the odd numbers, all numbers starting with 4 with a gap of 4 could be expressed as difference of 2 squares. For example 4 = (2)^2 - (0)^2, 40 = (11)^2 - (9)^2 etc.

 

Thus we have 250 odd numbers (in first 500 Natural numbers) + 125 numbers (multiples of 4) (in first 500 Natural numbers) that could be expressed as difference of squares.

 

250 + 125 = 375.

 

(375/500)*100 = 75%.

 

So 75% of the Numbers could be expressed as difference of two squares.

All numbers except those congruent to 2 mod 4 are representable as the difference of two perfect squares. These are just twice the odd numbers, so {2, 6, 10, 14, …}. There are 125 of these between 1 and 500, so 75% of the natural numbers less than 500 can be so written.

75%