HMMT 二月 2022 · ALGNT 赛 · 第 4 题

HMMT February 2022 — ALGNT Round — Problem 4

Compute the sum of all 2-digit prime numbers p such that there exists a prime number q for which 100 q + p is a perfect square.

解析

Compute the sum of all 2-digit prime numbers p such that there exists a prime number q for which 100 q + p is a perfect square. Proposed by: Sheldon Kieren Tan Answer: 179 Solution: All squares must end with 0, 1, 4, 5, 6, or 9, meaning that p must end with 1 and 9. Moreover, since all odd squares are 1 mod 4, we know that p must be 1 mod 4. This rules all primes 2 2 2 except for 41 , 61 , 29 , 89. Since 17 = 289 , 19 = 361 , 23 = 529, 89, 61, and 29 all work. To finish, we claim that 41 does not work. If 100 q + 41 were a square, then since all odd squares are 1 mod 8 we find that 4 q + 1 ≡ 1 (mod 8), implying that q is even. But 241 is not a square, contradiction. The final answer is 29 + 61 + 89 = 179.