1st ASU 1967 Problem 6 Find all integers x, y satisfying x 2 + x = y 4 + y 3 + y 2 + y.
The only solutions are x,y = -1,1-; 0,-1; -1,0; 0,0; -6,2; or 5,2. (y 2 + y/2 - 1/2)(y 2 + y/2 + 1/2) = y 4 + y 3 + 1/4 y 2 - 1/4 < y 4 + y 3 + y 2 + y except for -1 <= y <= -1/3. Also (y 2 + y/2)(y 2 + y/2 + 1) = y 4 + y 3 + 5/4 y 2 + y/2 which is greater than y 4 + y 3 + y 2 + y unless 0 <= y <= 2. But no integers are greater than y 2 + y/2 - 1/2 and less than y 2 + y/2. So the only possible solutions have y in the range -1 to 2. Checking these 4 cases, we find the solutions listed. 1st ASU 1967 (C) John Scholes jscholes@kalva.demon.co.uk 30 Sep 1998