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Math Help - Number Theory - Prime divisors

  1. #1
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    Number Theory - Prime divisors

    Can anyone start me off on this question?
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  2. #2
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    Given n primes, p_1, p_2, ..., p_n each of the form 8k + 7 and the number
    N = (p_1p_2...p_n)^2 - 2
    Show that N \equiv 7 (mod 8)

    Well, there are n primes. So the expansion
    N = (p_1p_2...p_n)^2 \equiv (7^n)^2 - 2 \equiv 7^{2n} - 2 \equiv 49^n - 2 (mod 8)

    Now 49 \equiv 1 (mod 8), so
    N \equiv 1 - 2 \equiv -1 \equiv 7 (mod 8)

    -Dan
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