PUMaC 2012 · 个人决赛（B 组） · 第 3 题

PUMaC 2012 — Individual Finals (Division B) — Problem 3

Find, with proof, all pairs ( x, y ) of integers satisfying the equation 3 x + 4 = 2 y . Please write complete, concise and clear proofs. Have fun! – PUMaC Problem Writers 1

解析

Find, with proof, all pairs ( x, y ) of integers satisfying the equation 3 x + 4 = 2 y . Solution: 3 3 3 This equation can be rewritten as (2 + x ) + (2 − x ) = (2 y ) . By Fermat’s last theorem, at least one of 2 + x , 2 − x , and y must be zero. The first case gives the solution ( − 2 , 2), the second case gives the solution (2 , 2), and the third case gives no solutions. Here is some motivation for this solution. We first note that x must be even, so let x = 2 k 2 3 (for some k ∈ Z ). Plugging this in and simplifying the equation gives us 6 k + 2 = y . Thus y 2 3 must be even as well, and letting y = 2 l gives 3 k + 1 = 4 l . Now we see that k must be odd, 2 3 so letting k = 2 m + 1 gives 3 m + 3 m + 1 = l . The left-hand side is easily recognizable as 3 3 3 3 3 ( m + 1) − m , so we may rewrite this equation as ( m + 1) = l + m . Thus either m = − 1, m = 0, or l = 0, and by considering these three cases we obtain the same two solutions. 2