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Math Help - Prime Numbers Congruence

  1. #1
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    Prime Numbers Congruence

    Theorem: If p is prime, and a + b = p - 1 for a, b positive integers, then a!*b! "is congruent to" (-1)^(b + 1) (mod p)

    a.) Illustrate two examples (w/ primes that are greater than 5) showing the theorem above.

    b.) Prove the above theorem.

    WORK:

    a.) Showing it:

    I'm assuming you pick a prime. So, let's pick 7 and 11.

    a + b = 7 - 1 = 6

    a!*b! "is congruent to" (-1)^(b + 1) (mod 7) ... not sure where to go from here.

    And similarly for 11:

    a + b = 11 - 1 = 10

    a!*b! "is congruent to" (-1)^(b + 1) (mod 11)

    And no clue how to do part b.

    Thanks!
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  2. #2
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    It reminds me of Gauss' Lemma. Here is the proof below.
    Attached Thumbnails Attached Thumbnails Prime Numbers Congruence-picture22.gif  
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