PUMaC 2021 · 数论（A 组） · 第 3 题

PUMaC 2021 — Number Theory (Division A) — Problem 3

Compute the number of ordered pairs of non-negative integers ( x, y ) which satisfy 2 2 x + y = 32045 . P e e e 3 1 2 k

解析

Compute the number of nonnegative integral ordered pairs ( x, y ) such that x + y = 32045. Proposed by: Nancy Xu Answer: 16 We can write 32045 = 5 · 13 · 17 · 29 = (1+2 i )(1 − 2 i )(2+3 i )(2 − 3 i )(1+4 i )(1 − 4 i )(2+5 i )(2 − 5 i ), 2 2 and from here we can write x + y = ( x − yi )( x + yi ) = 32045 by taking the product of one of 4 2 each of the conjugate pairs. There are 2 options for each conjugate pair for a total of = 8 2 to account for overcounting, but x and y can be swapped, so there are 16 nonnegative ordered pairs. P e e e 3 1 2 k