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Math Help - Divisibility of Mersenne numbers?

  1. #1
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    Divisibility of Mersenne numbers?

    Let M_n= 2^(n) - 1 be the n-th Mersenne number.
    a) Show that, if m|n, then M_m|M_n
    b) Show that, if m<n and m does not divide n, then GCD(M_n,M_m) = GCD(M_m,M_r) where r is the remainder of n upon division by m
    c) Let m,n be arbitrary natural numbers, and let d = GCD(m,n). Using the above results, show that GCD(M_n,M_m) = M_d

    So I figured out part a quite easily, by letting n=mk and using congruences, but I'm stuck on part b as im unsure whether this can by proved using only congruences or if I need to incorporate the Euclidean algorithm. Any ideas?
    Last edited by morbius27; April 23rd 2010 at 01:21 AM.
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  2. #2
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    Quote Originally Posted by morbius27 View Post
    Let M_n= 2^(n) - 1 be the n-th Mersenne number.
    a) Show that, if m|n, then M_m|M_n
    b) Show that, if m<n and m does not divide n, then GCD(M_n,M_m) = GCD(M_m,M_r) where r is the remainder of n upon division by m
    c) Let m,n be arbitrary natural numbers, and let d = GCD(m,n). Using the above results, show that GCD(M_n,M_m) = M_d

    So I figured out part a quite easily, by letting n=mk and using congruences, but I'm stuck on part b as im unsure whether this can by proved using only congruences or if I need to incorporate the Euclidean algorithm. Any ideas?
    M_d|M_n and M_d|M_m is easy.
    I would try to show GCD(M_n,M_m)=GCD(M_n,M_n-m) and then continue like in Euclidean algorithm. To me this seems as a standard and the most straightforward solution. (Maybe someone will come up with something more elegant.)
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