Track / Stage 5 / 371 of 400 #971 of 2000

Problem 971

AIME late

Number theory Difficulty 5.9 Prove it

14 Prove using elementary methods: There exist infinitely many triangular numbers that are also pentagonal numbers (i.e., numbers of the form $\frac{1}{2} k(3 k-1)$ , where $k$ is a positive integer).

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

The problem is to prove that the equation
$y(3 y-1)=x(x+1)$

has infinitely many positive integer solutions $x, y$ .
It is easy to verify: $(4 x+7 y+1)(12 x+21 y+2)-(7 x+12 y+1)(7 x+12 y+2)=$ $y(3 y-1)-x(x+1)$ , from which we immediately get: if there are positive integers $x, y$ satisfying (1), then the larger numbers
$x_{1}=7 x+12 y+10, y_{1}=4 x+7 y+1$

satisfy the equation $y_{1}\left(3 y_{1}-1\right)=x_{1}\left(x_{1}+1\right)$ . Since $x=y=1$ satisfies (1), the equation (1) has infinitely many positive integer solutions $x, y$ . In particular, from the solution $x-y=1$ and equation (2), we get the solution $x_{1}-20, y_{1}-12$ , and then the solution $x_{2}=285, y_{2}=165$ , and so on.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.