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Math Help - Congruences: "The characteristic p binomial theorem"

  1. #1
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    Congruences: "The characteristic p binomial theorem"

    a. Prove that (a+b)^p is congruent to a^p + b^p (mod p) for any integers a, b, provided p is prime.

    b. More generally, prove that if p is prime and q= p^n, where n is a natural number, then (a+b)^q is congruent to a^q + b^q (mod p) for any integers a,b. (Hint: Use Induction.)

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by meggnog View Post
    a. Prove that (a+b)^p is congruent to a^p + b^p (mod p) for any integers a, b, provided p is prime.

    b. More generally, prove that if p is prime and q= p^n, where n is a natural number, then (a+b)^q is congruent to a^q + b^q (mod p) for any integers a,b. (Hint: Use Induction.)

    Thanks in advance!

    (a+b)^q=\sum\limits_{k=0}^q\binom{q}{k}a^{q-k}{b^k} ... and now prove, using induction or whatever, that q\mid \binom{q}{k}\,\,\,\forall\, \,0 < k< q , q=p^n\,,\,\,p a prime.

    Tonio
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