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We have to prove that there exists infinitely many integers $a,b,c$ such that $a^2 + b^2 = c^2 + 3$ .

This looked like a very straight-forward question . I did some algebraic manipulations but couldn't reach the conclusion . Help me here

abkds
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1 Answers1

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You need $c^2 - a^2 = b^2 - 3 \iff (c-a)(c+a) = b^2- 3$

Let $b=3k$, then we need $(c-a)(c+a)= 3(3k^2-1)$, so let us try $c-a = 3, c+a = 3k^2-1$

Adding and subtracting, we get $c = \frac{3k^2+2}{2}, a = \frac{3k^2-4}{2}$, both would be integer if $k$ is even.

Hence one set of solutions is $a = 6k^2-2, b = 6k, c = 6k^2+1$, for integer $k$.

Macavity
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