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x^2+2=y^3 can only have one solution (if we're dealing in natural numbers), which was (5,3). Euler reproved the theorem since, like so many others of his, the proof was lost. I can't find the proof now, but I remember it had to do with complex numbers and relative primes.

I did some work with the theorem myself in my free time last summer, and I think I found a proof of it using infinite descent. I can't be certain my proof is correct since I haven't had the chance to have anyone review it, and I'm also not sure if anyone else has proven the theorem in a similar way. Does anyone know if it has? I'm especially proud because I think this is probably the way in which Fermat would have proven the theorem.