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Re: Puzzle - Law of small numbers

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  • Adam
    I think you can prove that Phil s example 103^16 is the smallest such number. Sigma(n) has factors (1+p+p^2+...+p^k) for p^k||n (
    Message 1 of 5 , Jan 10, 2007
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      I think you can prove that Phil's example 103^16 is the smallest
      such number. Sigma(n) has factors (1+p+p^2+...+p^k) for p^k||n (<-
      largest power of p that divides n). For sigma(n) to be 5 mod 6, all
      the factors have to be either 1 or 5 mod 6 (i.e., no factors that
      are 2,3,4,0 mod 6). By examination, for p=2 and p=3,
      (1+p+p^2+...+p^k) is never 0 mod 17 and 1 or 5 mod 6.

      For p=3, we get 1+3+3^2+...+3^k odd when k is even but then
      1+3+3^2+...+3^k=(3^(k+1)-1)/2 while the order of 3 mod 17 is 16 so 3^
      (k+1)-1 is divisible by 17 only when 16 divides k+1. It can not be
      true that k is even and 16 divides k+1.

      For p=2, we get 1+2+2^2+...+2^k is 1 mod 3 when k is even. But the
      order of 2 mod 17 is 8, so 2^(k+1)-1 == 0 mod 17 means (k+1) is even.

      The only other available primes are either 1 or 5 mod 6. If p is 5
      mod 6 and 1+p+...+p^k is either 1 or 5 mod 6, then k is even. If p
      is 1 mod 6 and 1+p+...+p^k is either 1 or 5 mod 6, then k is either
      4 or 6 mod 6.

      For k even, solve 1+x+...+x^k==0 mod 17 and find the first solution
      happens when k=16 and x==1 mod 17. The first prime that is 1 mod 17
      is 103. 103^16 is the first number that solve sigma(n)==17 mod 102.
    • jbrennen
      ... Yes, and you seemed to give a convincing argument. :) Did anyone notice that I basically gave away the key to finding Phil s example? I stated that
      Message 2 of 5 , Jan 10, 2007
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        --- In primenumbers@yahoogroups.com, "Adam" <a_math_guy@...> wrote:
        >
        > I think you can prove that Phil's example 103^16 is the smallest
        > such number.

        Yes, and you seemed to give a convincing argument. :)


        Did anyone notice that I basically gave away the key to finding
        Phil's example? I stated that solving sigma(N) == 5 (mod 6)
        was easily done by picking a prime p of the form 6*x+1 and
        then taking p^4, or p^(5-1).

        This is of course easily generalized...

        So to solve sigma(N) == 17 (mod 102), find a prime of the
        form 102*x+1. The first such is of course 103. And then
        set N = 103^(17-1) = 103^16.



        The tricky part is proving that 103^16 is in fact the minimal
        solution.

        My method for doing so:

        Note that N must be divisible by a prime power p^a which
        has sigma(p^a) divisible by 17 but not divisible by 2 or 3.

        Note also that sigma(p^a) = (p^(a+1)-1)/(p-1).

        In order for sigma(p^a) to be an odd number (not divisible
        by 2), we need to have either p or a be even.

        If p is even, it must be 2, and we would have:
        p^(a+1) == 1 (mod 17). It turns out that this is true
        when a == 7 (mod 8); however, when a == 7 (mod 8), we
        also have sigma(2^a) divisible by 3. So that doesn't
        work.

        So p must be odd and a must be even. In order for
        p^(a+1)-1 (the numerator of the expression above) to
        be divisible by 17, (a+1) must be divisible by the
        order of p modulo 17. Note that the order of p
        modulo 17 must divide evenly into 16 (a consequence
        of Fermat's Little Theorem). So we have an
        odd number (a+1) which must be divisible by a
        divisor of 16. Clearly the divisor (the order of
        p modulo 17) must be 1. This implies that the
        prime p is in fact of the form 17*x+1.

        And once we know that p == 1 (mod 17), we can see
        easily that a must be of the form 17*y-1.

        The smallest prime p of the form 17*x+1 is 103,
        and the smallest exponent a of the form 17*y-1 is 16.

        So the smallest prime power which has sigma(p^a)
        divisible by 17, but not by 2 or 3, is 103^16.

        So any solution N to the original puzzle must be
        divisible by a prime power >= 103^16.



        There are not a lot of ordered pairs (A,B) such
        that A > 2 and such that N = (B+1)^(A-1) is the
        smallest solution to sigma(N) == A (modulo B).

        I know of these three: (5,6), (5,30), and (17,102).

        There could be others, if anyone wants to try
        to find them. It's interesting that for those
        three I know of, A is always a Fermat prime.
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