14th Putnam 1954

------
 
 
Problem A7

Prove that the equation m2 + 3mn - 2n2 = 122 has no integral solutions.

 

Solution

Fairly easy.

If m, n is a solution, then 4m2 + 12mn - 8n2 = 488, so (2m + 3n)2 - 17n2 = 488, so (2m + 3n)2 = 12 (mod 17). But 12 is not a quadratic residue of 17 [check: 02, 12, 22, 32, 42, 52, 62, 72, 82 = 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.]

 


 

14th Putnam 1954

© John Scholes
jscholes@kalva.demon.co.uk
24 Nov 1999