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Math Help - Converse of Fermat's Little Theorem (particular case)

  1. #1
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    Converse of Fermat's Little Theorem (particular case)

    Let n = (6t+1)(12t+1)(18t+1) with t \in \{1,2,3...\} such that (6t+1),(12t+1) and (18t+1) are all prime numbers.

    Show that a^{n-1} = 1 (mod n) whenever hcf (a,n) = 1.
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  2. #2
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    Carmichael Numbers

    Carmichael Number -- from Wolfram MathWorld

    Let n=p_1p_2p_3 be of the following form:

    (6t+1)(12t+1)(18t+1)=1296t^3+396t^2+36t+1=36(36t^3  +11t^2+t)+1

    If each factor is prime, then p-1|n-1 for all p|n. By Korselt's criteria, this property makes n a Carmichael Number, which by definition, is any composite number that passes Fermat's test.
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