Difference between squares

Number Theory Level 4

How many positive integers from 1 to 2018 cannot be expressed as the difference between two perfect squares?

The answer is 505.

1 solution

Đức Minh Lê
Feb 21, 2018

If a positive integer $x$ can be written as a difference between two perfect squares, say $x=a^{2}-b^{2}$ , then $x=(a+b)(a-b)$ . It's obvious that $a+b$ and $a-b$ has the same parity.

If they are both even, then $x$ is divisible by $4$ .
If they are both odd, then $x$ is odd.

So $x$ cannot be expressed as a difference between two perfect squares if and only if $x$ has a remainder of $2$ when divided by $4$ .

From $1$ to $2018$ , such numbers are $2,6,10,...,2018$ . Thus our answer is $\frac{2018-2}{4}+1=\boxed{505}$

This argument has only shown one direction of your "if and only if" statement. You have shown that if $x \equiv 2$ mod $4,$ then it is not expressible. But you need to show that all $x \not\equiv 2$ mod $4$ are expressible (so that the answer is not larger than $505$ ).

Patrick Corn - 3 years, 2 months ago

Difference between squares

The answer is 505.

1 solution

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