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How to prove that the only integral solutions to the equation $$y^{2}=x^{3}-1$$ is $x=1, y=0$. I rewrote the equation as $y^{2}+1=x^{3}$ and then we can factorize $y^{2}+1$ as $$y^{2}+1 = (y+i) \cdot (y-i)$$ in $\mathbb{Z}[i]$. Next i claim that the factor's $y+i$ and $y-i$ are co-prime. But i am not able to show this. Any help would be useful. Moreover, i would also like to see different proofs of this question.

Extending Consider the equation $$y^{a}=x^{b}-1$$ where $a,b \in \mathbb{Z}$ and $(a,b)=1$ and $a < b$. Then is there any result which states about the nature of the solution to this equation.

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The second question: en.wikipedia.org/wiki/Catalan's_conjecture –  Qiaochu Yuan Oct 3 '10 at 10:37
    
@Chandru1 : sorry, could you please explain to me that factorization (y+i)(y-i)? I don't know what it means. –  Weltschmerz Oct 3 '10 at 13:56
    
He's saying that integers of the form $y^2+1$ can be represented as a product of two Gaussian integers, $y+i$ and $y-i$, where $i$ is the usual imaginary unit. –  J. M. Oct 3 '10 at 15:10
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3 Answers

up vote 5 down vote accepted

Alon Amit's answer is not right, I believe.

$2$ is not prime in the Gaussian integers! $(1+i)$ divides $2$.

The argument upto the fact that the prime $p$ divides $y+i$, $y-i$ and $2i$ is correct.

Since $p|2i$ we consider $p = 1+i$. Now any multiple of $1+i$ is of the form $2x + i2y$ or $2x+1 + i(2y+1)$. Since $y$ is even, $y+i$ cannot be divisible by $1+i$ and thus $y+i$ and $y-i$ are co-prime.

Setting $y+i$ to be a perfect cube (upto units) easily gives us $y=0$ and so $x=1, y=0$ is the only solution.

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If $x$ is even then $x^3$ is divisible by 4, so $x^3-1 \equiv 3 \pmod 4$ and this cannot be a square. Thus $x$ is odd and $y$ is even.

Now, if a prime $p$ divides both $y+i$ and $y-i$ then it divides their difference $2i$. Thus $p=2$ (up to units), but then $p$ divides $y$. That's impossible since it divides $y+i$ as well.

EDIT: I was too hasty in writing this, as pointed out by Moron. The prime 2 ramifies in $\mathbb{Z}[i]$, and is (up to units) the square of $1+i$. Sorry for the confusion.

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Is it really necessary to use complex numbers here? Does this sound reasonable to you or is there a flaw? According to the 7-period for n^2 and n^3-1 (mod 7) LHS=RHS requires both to be 0 mod(7), so that for some n and k y^2=n*7^2 must equal k* 7^3 -1 and therefore 7^2(n-7k)=-1 , impossible, since 7^2>1. I note the table format doesn't come out right - you have to do it yourselves.

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Welcome to MSE! I realize you don't yet have enough reputation, but this is better left as a comment as it is not explicitly answering the question and posing another one. Regards –  Amzoti Apr 9 '13 at 12:52
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