Math Help - 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 has only and as its solutions. By far the best way to prove this is over the unique factorization domain . Because you can factor the left hand side as .

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