HMMT 二月 2005 · 冲刺赛 · 第 35 题

英文解析

35. Let p = 2 − 1, the largest prime currently known. For how many positive integersc do the quadratics ± x ± px ± c all have rational roots?2
    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 ). Butp ≡ 3 (mod 4) and is therefore not the sum of two squares. This implies that p is notthe hypotenuse of any Pythagorean triple, so the answer is 0.
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