The Sandwich Number

I came across this interesting fact while browsing one of the mathematical books. It is the sandwich number as termed by Fermat. A sandwich number is one, which is the only number which lies in between a perfect-square and a perfect-cube. One example of it is the number 26. It lies between 25(which is 5 squared) and 27(which is 3 cubed). The astonishing thing proposed by Fermat was 26 is the ONLY sandwich number!! And he even put forward a proof in his usual style(scribbling in margins).

Although this fact appears simple, the proof might either be much complicated or a tricky one, as is the case with many number theory problems. If anyone comes across a proof to this problem, please enlighten me!

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