Same Differences

Given the positive integers, $x$, $y$, and $z$, are consecutive terms of an arithmetic progression, the least value of the positive integer, $n$, for which the equation, $x^2 – y^2 – z^2 = n$, has exactly two solutions is $n = 27$:
$$34^2 – 27^2 – 20^2 = 12^2 – 9^2 – 6^2 = 27.$$
It turns out that $n = 1155$ is the least value which has exactly ten solutions.
How many values of $n$ less than one million have exactly ten distinct solutions?

We note that $(x^2-y^2)$ is a simple difference of squares identity equal to $(x+y)(x-y)$, and set it equal to $(m)(n)$. Thus, the problem can be restated as follows:

Find the number of positive integers less than one million which can be expressed as the product of two integers $(m)(n)$ such that $m – n = z$ for exact ten positive integral $z$s.

Let $M$ be the set of values that $m$ can take, and $z$ be the corresponding difference $m – n$.

Given $m$ and $z$, we can figure out $n$ as $n = m – z$. Now, it should be clear that $z$ can range from $1$ to $m/2$ (since $m – n$ must be positive and less than or equal to half of $m$). Hence, for every $m$ in set $M$, there will be $\lfloor \frac{m}{2} \rfloor$ corresponding $z$s.

We want to find the count of such $m$’s between 1 and $10^6$ for which there are exactly 10 corresponding $z$’s or $\lfloor \frac{m}{2} \rfloor = 10$. This comes out to be 19 (for $m$ = 21, 22, 23, …, 39).

Note that we didn’t consider whether $x$, $y$ and $z$ are integers themselves or not! But it is already pre-defined in the question that they have to be integers. This needs $m$ and $n$ to be both even, or both odd.
For two numbers $m$ and $n$ to have the same parity but their difference ($z$) be an odd integer (as is necessary for our case, because all our defined $z$’s lie between 10 and 20), both $m$ and $n$ should be odd numbers.

So, consider only odd numbers from the 19 values of $m$ that we have.
Now, answer = all odd $m$s between 21 and 39. Hence, the answer is 10.

So, there are 10 values of $n$ less than one million have exactly ten distinct solutions.

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