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Math Help - Carmichael Numbers

  1. #1
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    Carmichael Numbers

    1.) Show for every integer of this form: (6m + 1)(12m + 1)(18m + 1), with m being a pos. integer such that 6m + 1, 12m + 1, 18m + 1 are all primes, is a Carmichael number

    2.) From part #1, show why 294409 = 37*73*109 is a Carmichael number

    3.) Determine the least pos. residue of 765^(1472043) (mod 294409)

    WORK:

    From my number thry book:

    A composite integer n that satisfies b^(n-1) = 1 (mod n) for all pos. integers b with (b,n) = 1 is called a Carmichael number or an absolute pseudoprime.

    And a THM about thsee numbers:

    If n = q_1q_2...q_k, where q_j are distinct primes that satisfy (q_j - 1) | (n - 1) for all j and k > 2, then n is a Carmichael number.
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  2. #2
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    Klicken sie Heir

    And follow the same approach.
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