HMMT 十一月 2008 · 冲刺赛 · 第 31 题

英文解析

31. [ 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.