Solving equations depending on $z$

Question

For witch real number $z$, are there positive integers $a$, $b$ and $c$, that fulfill this system of equations? $(a,b,c)∈ℕ$

$a+b+c=57$

$a^2+b^2−c^2=z$

$z·c=2017$

I need to determine all variables $(a, b, c)$ depending on $z$. Any hints?

Answer 1

I have found a solution for my problem.

There are no numbers that fulfill the equation. So (a,b,c) ∈ ℕ

$z*c=2017$

2017 is a Prime number. If

$z ≠ 1 ⋁ 2017$

then $c$ will logically be an odd number, because else if z is not 1 or 2017 we'll have:

$2017/z = c$ (≠ even)

So $z$ has to be definitely 1 or 2017.

So we have case one: c = 1 if z is 2017 and case two: c = 2017 if z is 1.

Case one: $(c=1;z=2017)$

1.) $a+b-1 = 57$ 2.) $a^2 +b^2 -c^2= 2017$

If we divide 2.) by 1.) we get:

1.) $a + b + 1 = 57 => a+b = 56$

$a^2 + b^2 -1^2 = 2017 => a^2 + b^2 = 2018$

2.) / 1.) $(a^2 +b^2) / (a+b) = 2018 / 56$

This makes no sense... ↯

Case two: $(c=2017;z=1)$

1.) $a+b +2017 = 57$

2.) $a^2 + b^2 - 2017^2 = 1$

We end up with:

$(a^2 + b^2) / (a+b) = (2017^2) / (57-2017)$

I'm not 100% sure but i guess i'm right. If not, pls comment.

Thx.