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Math Help - another fermat's little theorem question

  1. #1
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    another fermat's little theorem question

    prove that 5n+3, where n is a prime, can never be a perfect square
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  2. #2
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    Hello,
    Quote Originally Posted by lmasud View Post
    prove that 5n+3, where n is a prime, can never be a perfect square
    Hmmm I don't see how to apply Fermat's little theorem here o.O

    So you want to prove that there is no x such that x^2=5n+3, that is x^2\equiv 3 (\bmod 5)
    If x=0(mod5), then x≤=0(mod5)
    If x=1(mod5), then x≤=1(mod5)
    If x=2(mod5), then x≤=4(mod5)
    If x=3(mod5), then x≤=4(mod5)
    If x=4(mod5), then x≤=1(mod5)

    thus it x doesn't exist.
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    thx...im wasnt sure if i needed to apply FLT, but thats what we're doing in class, so i thought it had something to do with it
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    Hmm if you want to apply FLT, it's actually possible.

    Assume there is x such that x^2\equiv 3(\bmod 5)
    Square it :
    x^4\equiv 9(\bmod 5) \equiv 4(\bmod 5)

    But we know, by FLT, that for any x coprime with 5, x^4\equiv 1(\bmod 5)
    And if it is not coprime with 5, it means that it's a multiple of 5. In which case, x^4\equiv 0(\bmod 5)

    So we have a contradiction and there is no x such that x^2\equiv 3(\bmod 5)

    Is it more similar to what you're doing in class ?
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