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How to prove that the equation $$x^2-3y^2=17$$ has no integer solutions? Can you help me?

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@Ayman Hourieh Thanks a lot! –  Dao yi Peng Jan 13 '13 at 13:14

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up vote 3 down vote accepted

$$x^2-3y^2=17\implies x^2\equiv2\pmod 3$$

but $x$ can be $\equiv 0,\pm1\pmod 3\implies x^2\equiv0,1\pmod 3$

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@bhattachaejee Thank you very much! –  Dao yi Peng Jan 13 '13 at 13:15
    
@DaoyiPeng, my pleasure. Don't miss the related question. –  lab bhattacharjee Jan 13 '13 at 13:16
    
bhattachaejee OK! I need to see some Pell's eqution. –  Dao yi Peng Jan 13 '13 at 13:18
    
@labbhattacharjee could you explain to me the step x can be $0, +- 1$? I'm not sure how that proves the answer.. –  user128914 Mar 7 at 1:08
    
@user128914, what are the possible remainders when divided by $3$? –  lab bhattacharjee Mar 7 at 3:20

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