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Problem 567

AMC 12 late, AIME early

Algebra Difficulty 5.0 Find the answer

Example 1 Find all integer solutions to the indeterminate equation $x^{2}=y^{2}+2 y+13$ .

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Your answer

Official solution

The original equation can be transformed into
$x^{2}-(y+1)^{2}=12 \text {. }$

By the difference of squares formula, we get
$(x+y+1)(x-y-1)=12 \text {. }$

Since $x+y+1$ and $x-y-1$ have the same parity, and 12 is an even number, we have
$\begin{array}{l} (x+y+1, x-y-1) \\ =(2,6),(6,2),(-2,-6),(-6,-2) \\ \Rightarrow(x, y)=(4,-3),(4,1),(-4,1), \\ (-4,-3) . \end{array}$

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