# does $a^2-51b^2=\mp 6$ have a solution for integers?

does $a^2-51b^2=\mp 6$ have a solution for integers?

I have tried for many modulos, but could not get much out of them.

• Are you familiar with quadratic residues? – Daniel Fischer Dec 24 '13 at 21:00
• yes. would mod 5 work? – 104078 Dec 24 '13 at 21:02
• The key, incidentally, is to pick a modulus that simplifies the problem; since $51=3\cdot17$, those are the places to start. – Steven Stadnicki Dec 25 '13 at 1:36
• – lab bhattacharjee Dec 25 '13 at 3:27

## 2 Answers

A solution to $a^2 - 51 b^2 = \pm 6$ would in particular be a solution to the congruence

$$a^2 \equiv \pm 6 \pmod{17}.$$

But neither $6$ nor $11$ is a quadratic residue modulo $17$ - the quadratic residues are $1,2,4,8,9,13,15,16$, so there is no solution.

• beat me to it! :) – frogeyedpeas Dec 24 '13 at 21:04

$\rm mod\ 17\!:\ a^2\!\equiv \pm 6\Rightarrow a^4\!\equiv 6^2\!\equiv 2\Rightarrow a^8\!\equiv 4\Rightarrow a^{16}\!\equiv -1\,$ contra little Fermat. $\$ QED