Prove that \(25 x^2 6x 95 = y^2y\quad \) has no positive integer solutions \((x,y)\).

03262010, 11:51 AM
(This post was last modified: 04022010 03:11 PM by elim.)
Post: #1




Prove that \(25 x^2 6x 95 = y^2y\quad \) has no positive integer solutions \((x,y)\).
Please try in number theory's way


04022010, 03:24 PM
Post: #2




RE: Prove that \(25 x^2 6x 95 = y^2y\quad \) has no positive integer solutions \((x,y)\).
Solve \(y \quad (y>0)\) to get \(y=(1/2)(1+\sqrt{100x^224x379})\)
So it's necessary to have \(x,n \in \mathbf{N}^+\) such that \(100x^224x379 = n^2\) thus \(x = (6+\sqrt{25n^2+9511})/50\) and so \(25n^2+9511\) is a perfect square hence \(25n^2+9511=m^2\) for some \(m \in \mathbf{N}^+\). Now \(9511\) is prime and \((m+5n)(m5n)=9511\), we see that \(5n+m=9511,\quad m=5n+1\) and so \(n=951\). But then \(x\) would not be an integer. We thus get a contradiction. 

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