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Thread: Lagrange Theorem etc.

  1. #1
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    Lagrange Theorem etc.

    Define a sequence $\displaystyle \{a_{i} \} $ by $\displaystyle a_1 = 4 $ and $\displaystyle a_{i+1} = 4^{a_{i}} $ for $\displaystyle i \geq 1 $. Which integers between $\displaystyle 00 $ and $\displaystyle 99 $ inclusive occur as the last two digits in the decimal expansion of infinitely many $\displaystyle a_{i} $.

    It would take a long time to do this by brute force. What exactly does this mean (this was a hint that was given)?: If $\displaystyle 4 $ does not divide $\displaystyle n $, then $\displaystyle 4^{a} \mod n $ is determined by $\displaystyle a \mod \phi(n) $. So for example, $\displaystyle 4^{a}\mod15 $ is determined by $\displaystyle a \mod \phi(15) $.
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  2. #2
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    I think you are expected to know that if $\displaystyle gcd(a,m) =1$ then $\displaystyle a^{\phi(m)}\equiv 1 \quad (mod m)$, where phi is the Euler phi function

    I suspect you are capable of sussing things out from there.
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