HMMT 二月 2002 · 代数 · 第 2 题
HMMT February 2002 — Algebra — Problem 2
题目详情
- p and q are primes such that the numbers p + q and p + 7 q are both squares. Find the value of p .
解析
- p and q are primes such that the numbers p + q and p + 7 q are both squares. Find the 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 taken 2 to be positive; then we must have y − x = 2 , y + x = 6 ⇒ x = 2, so p + 2 = 2 = 4 ⇒ p = 2 also.