Variables and square number problem

Question

I'm having issues with this problem.

$N + 2000$ is an square number. $N - 17$ is also an square number. Find $N$.

Answer 1

Hint:

Let,

$$N+2000=a^2$$

$$N-17=b^2$$

So that,

$$(N+2000)-(N-17)=a^2-b^2$$

$$2017=(a-b)(a+b)$$

$2017$ is prime so there isn't really that many ways to break it apart into multiplication of integers.

Answer 2

$N+2000=p^{2}$ (1)

$N-17=q^{2}$ (2)

Substract (2) from (1)

$N+2000-(N-17)=p^{2}-q^{2}$

$2017=(p-q)(p+q)$

2017 is a prime, thus:

$p-q=1$

$p+q=2017$

Solving the system of equations gives you p=1009, q=1008. Now use p in (1), or q in (2) $N+2000=1009^{2}$ And the desired number is 1016081.