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Thread: Congruence

  1. #1
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    Congruence

    I need help solving congruence: a^{p-1}+a^{p-2}+...+a^2+a+1=0 (mod p), p is prime.
    I know that I can use fermat little theorem (a^{p-1}=1 (mod p)) and I know that \Z_p does not have zeros.
    I have tried with some primes but can't see the similarity.
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  2. #2
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    Re: Congruence

    I assume you mean solve for a? But the only solution is a=1:
    $$a^{p-1}+a^{p-2}+\cdots+a+1\equiv0\,(\text{mod }p)$$
    $$(a^{p-1}+a^{p-2}+\cdots+a+1)(a-1)\equiv0\,(\text{mod }p)$$
    $$a^{p}-1\equiv0\,(\text{mod }p)$$
    $$a\equiv a^{p}\equiv1\,(\text{mod }p)$$
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