How many integers $k$ are there between $1$ and $1000$ inclusive that can be expressed as a difference of two squared integers?

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  1. Factorise $a^3-b^3.$
  2. If $a$ is divisible by $2$ and $b$ is divisible by $4,$ would $ab$ be divisible by $8?$
  3. If $m$ is divisible by $2$ and $4,$ would $m$ be divisible by $8?$

Have you expressed the question statement in mathematical terms and simplified it?

Pay close attention to the two divisors of $k.$

… in particular, their odd-even parity.

Could they be odd or even independently? What are the possible combinations?

You should find that they have to be both odd or both even, as a necessary condition. Is it a sufficient condition?

Try giving an expression for $a$ and $b$ in terms of $k$ for each scenario.

Let’s assume that $a^2-b^2=k,$ for some $k \in \{1,2,3,\ldots,1000\}.$ Now, we can write $(a+b)(a-b) = k.$ Notice that $(a+b)$ and $(a-b)$ are both odd or both even. We can look at the two cases separately.

When $(a+b)$ and $(a-b)$ are both odd, $k$ must be an odd integer with two odd divisors. Every odd integer between $1$ and $1000$ has at least two odd divisors that are $1$ and itself. Hence, when $k$ is odd, we can take $a=\frac{(k+1)}{2}$ and $b=\frac{(k-1)}{2}.$ Indeed, we can show that any odd number can be written as the difference between two perfect squares: $2n+1=(n+1)^2-n^2$ for $n \in \mathbb{N}.$

When $(a+b)$ and $(a-b)$ are both even, $k$ must be divisible by $4.$ We can take $a=\frac{k}{4}+1$ and $b= \frac{k}{4}-1.$

Since there are $250$ integers divisible by $4$ and $500$ odd integers between $1$ and $1000,$ the answer is $750.$

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