Prove that there are infinitely many positive integers m, such that n4 + m is not prime for any positive integer n.
Solution
n4 + 4 r4 = (n2 + 2rn + 2r2)(n2 - 2rn + 2r2). Clearly the first factor is greater than 1, the second factor is (n - r)2 + r2, which is also greater than 1 for r greater than 1. So we may take m = 4 r4 for any r greater than 1.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
5 Oct 1998
Last corrected/updated 5 Oct 1998