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Math Help - Lagrange Theorem etc.

  1. #1
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    Lagrange Theorem etc.

    Define a sequence  \{a_{i} \} by  a_1 = 4 and  a_{i+1} = 4^{a_{i}} for  i \geq 1 . Which integers between  00 and  99 inclusive occur as the last two digits in the decimal expansion of infinitely many  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  4 does not divide  n , then  4^{a} \mod n is determined by  a \mod \phi(n) . So for example,  4^{a}\mod15 is determined by  a \mod \phi(15) .
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  2. #2
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    I think you are expected to know that if gcd(a,m) =1 then 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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