Prove that the equation m^{2} + 3mn - 2n^{2} = 122 has no integral solutions.

**Solution**

*Fairly easy.*

If m, n is a solution, then 4m^{2} + 12mn - 8n^{2} = 488, so (2m + 3n)^{2} - 17n^{2} = 488, so (2m + 3n)^{2} = 12 (mod 17). But 12 is not a quadratic residue of 17 [check: 0^{2}, 1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, 6^{2}, 7^{2}, 8^{2} = 0, 1, 4, 9, 16, 8, 2, 15, 13 (mod 17)].

[How do we think of this? Well, completing the square is a fairly natural procedure. Having done it, we wonder if 488 is a quadratic residue and find it isn't.]

© John Scholes

jscholes@kalva.demon.co.uk

24 Nov 1999