25th Brazil 2003 Problem A1 Find the smallest positive prime that divides n 2 + 5n + 23 for some integer n.
17 Solution Put f(n) = n 2 + 5n + 23. Calculating f(n) for n = -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4 we get 47, 37, 29, 23, 19, 17, 17, 19, 23, 29, 37, 47, 59. Note that f(-n-5) = f(n), so this only involves calculating 7 terms. None of these are divisible by any of 2, 3, 5, 7, 11, 13. But all possible values mod 13 are assumed by f(n) for any 13 consecutive integers, similarly for 11, 7 etc. So f(n) is not divisible by 2, 3, 5, 7, 11, 13 for any n. On the other hand f(-2) = 17. 25th Brazil 2003 © John Scholes jscholes@kalva.demon.co.uk 16 January 2004 Last corrected/updated 16 Jan 04