HMMT 二月 2002 · 代数 · 第 2 题

英文解析

2. p and q are primes such that the numbers p + q and p + 7 q are both squares. Findthe value of p .
   2 2 2 2
   Solution: 2 . Writing x = p + q, y = p + 7 q , we have 6 q = y − x = ( y − x )( y + x ).
   Since 6 q is even, one of the factors y − x, y + x is even, and then the other is as well; thus
   6 q is divisible by 4 ⇒ q is even ⇒ q = 2 and 6 q = 12. We may assume x, y are both takento be positive; then we must have y − x = 2 , y + x = 6 ⇒ x = 2, so p + 2 = 2 = 4 ⇒ p = 22
   also.