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Math Help - Show co-prime...

  1. #1
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    Show co-prime...

    Suppose that a,b\in\mathbb{N}^+,\ \gcd(a,b) = 1 and p is an odd prime.

    Show that \gcd\left(a+b,\frac{a^p+b^p}{a+b}\right) \in\{1,p\}
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  2. #2
    Super Member PaulRS's Avatar
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    Re: Show co-prime...

    Note that a^p + b^p = (a + b) \cdot ( a^{p-1} - a^{p-2} b + a^{p-3} b^2  ... + b^{p-1})
    (because, more generally x^n - y^n = (x-y)\cdot (x^{n-1}y^0 + x^{n-2}y^1 + ... + x^1 y^{n-2} + x^0y^{n-1}) now substitute: x = a, y = -b, n = p )

    So then \frac{a^p + b^p}{a+b} =  a^{p-1} - a^{p-2} b + a^{p-3} b^2  ... + b^{p-1} , but now  a^{p-1} - a^{p-2} b + a^{p-3} b^2  ... + b^{p-1} \equiv a^{p-1} - a^{p-2}\cdot (-a) + ... + (-a)^{p-1} (\bmod a + b) since a\equiv - b(\bmod.a+b)

    That is \frac{a^p + b^p}{a+b} \equiv p\cdot a^{p-1} (\bmod. a + b) \dots
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  3. #3
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    Re: Show co-prime...

    Thanks a lot PaulRS!
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