**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.

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