54th Kürschák 1953

n and d are positive integers such that d divides 2n². Prove that n² + d cannot be a square.

Solution

Put 2n² = dk. Suppoe that for some integer m, we have m² = n² + d, so m²k² = n²k² + 2n²k = n²(k² + 2k). So (mk/n)² = k² + 2^k, an integer. Hence mk/n is an integer. So k² + 2k is a square. But that is impossible because it is between the adjacent squares k² and (k+1)².

54th Kürschák 1953

© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03