HMMT 二月 2005 · 冲刺赛 · 第 35 题

HMMT February 2005 — Guts Round — Problem 35

[12] Let p = 2 − 1, the largest prime currently known. For how many positive 2 integers c do each of the quadratics ± x ± px ± c have rational roots?

解析

Let p = 2 − 1, the largest prime currently known. For how many positive integers 2 c do the quadratics ± x ± px ± c all have rational roots? Solution: 0 2 2 This is equivalent to both discriminants p ± 4 c being squares. In other words, p must 2 2 be the average of two squares a and b . Note that a and b must have the same parity, 2 2 a + b a − b a + b 2 2 2 and that ( ) + ( ) = = p . Therefore, p must be the hypotenuse in a 2 2 2 2 2 2 2 Pythagorean triple. Such triples are parametrized by k ( m − n , 2 mn, m + n ). But p ≡ 3 (mod 4) and is therefore not the sum of two squares. This implies that p is not the hypotenuse of any Pythagorean triple, so the answer is 0. 13