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Math Help - Number theory challenge

  1. #1
    Super Member Bacterius's Avatar
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    Number theory challenge

    Hi all, here is a little challenge for you all, involving number theory.

    Let a_{n + 1} \equiv {a_n}^3 \pmod{113}, a_1 being the initial term.

    If I tell you that a_4 = 26, can you find a_1 ?
    Good luck

    Please don't just try all possible values ...
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  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
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    Quote Originally Posted by Bacterius View Post
    Hi all, here is a little challenge for you all, involving number theory.



    Good luck

    Please don't just try all possible values ...

    From Fermat's Little theorem, we have a_1^{112} = 1 \mod 113
    Following the recursion gives a_1^{27} = 26 \mod 113. But a_1^{27} = 26 \mod 113 \implies a_1^{108} = 26^4 \mod 113 \implies a_1^{4} = 26^{-4} \mod 113
    So definitely a_1 = 13 \mod 113 is one possible answer. I dont know whether there are more.
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  3. #3
    Super Member Bacterius's Avatar
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    Hello,

    a_1 = 13

    a_2 = 50

    a_3 = 22

    a_4 = 26

    That was exactly the one I was thinking of Would you have taken the same time if I had taken a five-digit modulus ?

    Well done anyway
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