6th Iberoamerican 1991 Problem 5 Let p(m, n) be the polynomial 2m 2 - 6mn + 5n 2 . The range of p is the set of all integers k such that k = p(m, n) for some integers m, n. Find which members of {1, 2, ... , 100} are in the range of p. Show that if h and k are in the range of p, then so is hk.
1,2,4,5,8,9,10,13,16,17,18, 20,25,26,29,32,34,36,37, 40,41,45,49,50,52,53,58, 61,64,65,68,72,73,74, 80,81,82,85,89,90,97,98,100 Solution We have p(m,n) = (m-2n) 2 + (m-n) 2 , so p(2a-b,a-b) = a 2 + b 2 . Hence the range of p is just the sums of two squares. (a 2 + b 2 )(c 2 + d 2 ) = (ac - bd) 2 + (ad + bc) 2 , which establishes that if h and k are in the range, then so is hk. Thanks to Johann Peter Gustav 6th Ibero 1991 © John Scholes jscholes@kalva.demon.co.uk 12 January 2004 Last corrected/updated 12 Jan 04