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Math Help - proof that 26 is the only number between a cubed and a squared number

  1. #1
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    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
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  2. #2
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    It comes done to proving that x^2 + 2 = y^3 has only x=5 and y=3 as its solutions. By far the best way to prove this is over the unique factorization domain \mathbb{Z}[\sqrt{-2}] = \{ a+bi\sqrt{2}|a,b\in\mathbb{Z} \}. Because you can factor the left hand side as (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.
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  3. #3
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    Here's a link on the matter. Interesting little algebraic number theory problem.

    Math Forum - Ask Dr. Math
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  4. #4
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    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
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