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HMMT 十一月 2008 · 冲刺赛 · 第 31 题

HMMT November 2008 — Guts Round — Problem 31

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 15 ] Find the sum of all primes p for which there exists a prime q such that p + pq + q is a square.
解析
  1. [ 15 ] Find the sum of all primes p for which there exists a prime q such that p + pq + q is a square. 2 2 2 2 2 Answer: 8 3 and 5 both work, because 3 + 3 · 5 + 5 = 49. Now, say p + pq + q = k , for a 2 2 positive integer k . Then ( p + q ) − k = pq , or: ( p + q + k )( p + q − k ) = pq Since p + q + k is a divisor of pq , and it is greater than p and q , p + q + k = pq . Then p + q − k = 1. So: 2 p + 2 q = pq + 1 ⇔ pq − 2 p − 2 q + 4 = 3 ⇔ ( p − 2)( q − 2) = 3 This shows that one of p and q is 3 and the other is 5.