Thread: proof that 26 is the only number between a cubed and a squared number

I was reading Singh's Fermats Last Theorem, and it mentioned that fermat was the first person to prove that 26 is the only number between a cubed and a squared number. I have been unable to prove this, so i thought I'd ask, apologies if its been asked before.... cheers

It comes done to proving that $\displaystyle x^2 + 2 = y^3$ has only $\displaystyle x=5$ and $\displaystyle y=3$ as its solutions. By far the best way to prove this is over the unique factorization domain $\displaystyle \mathbb{Z}[\sqrt{-2}] = \{ a+bi\sqrt{2}|a,b\in\mathbb{Z} \}$. Because you can factor the left hand side as $\displaystyle (x+i\sqrt{2})(x-i\sqrt{2}) = y^3$.

Of course Fermat did not in any way prove this using the idea above. Unique factorization was only developed about 100 years ago by Kummer/Kronecker. I have no idea how Fermat, or Euler who later solved it also, did it by using elementary results.

Here's a link on the matter. Interesting little algebraic number theory problem.

Math Forum - Ask Dr. Math

that maths is beyond me, i think i was decieved by the seeminly simple nature of the problem. But i will look into some of the areas I don't understand, thanks