## Sunday, March 19, 2006

### Lotta Solutions. Topic: Number Theory. Level: Olympiad.

Problem: (1996 Italy - #2) Prove that the equation $a^2+b^2 = c^2+3$ has infinitely many integer solutions $\{a,b,c\}$.

Solution: Let $a$ be any even integer. Setting $b = \frac{a^2-4}{2}$ and $c = \frac{a^2-2}{2}$, we have

$a^2+b^2 = a^2+\left(\frac{a^2-4}{2}\right)^2 = \frac{a^4-4a^2+16}{4} = \left(\frac{a^2-2}{2}\right)^2+3 = c^2+3$,

as desired. Hence there are infinitely many integer solutions. QED.

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Comment: With these problems it's a good idea to find solutions for some small values. I found the solutions $(2,0,1)$, $(4,6,7)$, and $(6,16,17)$ from which I guessed that if we set $b = n$, $c = n+1$ for some integer $n$ we might be able to generate some solutions. After writing $n$ in terms of $a$, this was easy.

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Practice Problem: Find all solutions to the equation $x^3+y^3 = z^2$ in the positive integers.