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Chris Smyth

  1. The divisibility of a^n-b^n by powers of n.

    Втр, 09/15/2009 - 11:27 — SomNambul
    Authors: Chris Smyth
    Subjects: Number Theory
    Abstract

    For given integers a,b, and j at least 1 we determine the set of integers n
    for which a^n-b^n is divisible by n^j. For j=1,2, this set is usually infinite;
    we find explicitly the exceptional cases for which a,b the set is finite. For
    j=2, we use Zsigmondy's Theorem for this. For j at least 3 and gcd(a,b)=1, the
    set is probably always finite; this seems difficult to prove, however.

    We also show that determination of the set of integers n for which a^n+b^n is
    divisible by n^j can be reduced to that of the above set.

  2. The divisibility of a^n-b^n by powers of n.

    Втр, 09/15/2009 - 11:27 — SomNambul
    Authors: Chris Smyth
    Subjects: Number Theory
    Abstract

    For given integers a,b, and j at least 1 we determine the set of integers n
    for which a^n-b^n is divisible by n^j. For j=1,2, this set is usually infinite;
    we find explicitly the exceptional cases for which a,b the set is finite. For
    j=2, we use Zsigmondy's Theorem for this. For j at least 3 and gcd(a,b)=1, the
    set is probably always finite; this seems difficult to prove, however.

    We also show that determination of the set of integers n for which a^n+b^n is
    divisible by n^j can be reduced to that of the above set.

  3. The terms in Lucas sequences divisible by their indices.

    Чт, 08/27/2009 - 13:23 — SomNambul
    Authors: Chris Smyth
    Subjects: Number Theory
    Abstract

    For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as
    usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are
    either integers or conjugate quadratic integers, we describe the set of indices
    n for which n divides u_n and also the set of indices n for which n divides
    v_n. Building on earlier work, particularly that of Somer, we show that the
    numbers in these sets can be written as a product of a so-called basic number,
    which can only be 1, 6 or 12, and particular primes, which are described
    explicitly.

  4. The terms in Lucas sequences divisible by their indices.

    Чт, 08/27/2009 - 13:23 — SomNambul
    Authors: Chris Smyth
    Subjects: Number Theory
    Abstract

    For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as
    usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are
    either integers or conjugate quadratic integers, we describe the set of indices
    n for which n divides u_n and also the set of indices n for which n divides
    v_n. Building on earlier work, particularly that of Somer, we show that the
    numbers in these sets can be written as a product of a so-called basic number,
    which can only be 1, 6 or 12, and particular primes, which are described
    explicitly.

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