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The primeness of even n compared to odd n in k*2^n+/-1

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  • robert44444uk
    Recently I announced the discovery of a super-prime power series of the form k*2^n-1, n variable, k fixed, with over 200 primes to the Yahoo Primeform group.
    Message 1 of 2 , Aug 19, 2013
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      Recently I announced the discovery of a super-prime power series of the form k*2^n-1, n variable, k fixed, with over 200 primes to the Yahoo Primeform group.

      http://tech.groups.yahoo.com/group/primeform/message/11407

      An analysis of the primes discovered to date shows that there are significantly more even n in the list of primes (119) than odd (94). The complete list of the 213 primes to date is at:

      http://www.mersenneforum.org/showthread.php?t=18407 message 5.

      I have been fascinated for some while at the ability of k.4^n+/-1 to generate long Cunningham chains, and I am wondering if the predominance of even members of the super-prime power series is connected to the fact that no prime p is p-1 modulo a square base, 4 being 2^2 of course.

      I had done some initial work on generalised Cunningham chains back in 2007, and - found good length chains quickly for b=4 and 9, see

      http://www.mersenneforum.org/showthread.php?t=9021

      Comments, remarks, brickbats welcome

      Robert
    • thomas_ritschel
      ... The occurence of more even exponents in the list of primes is a direct result of the distrubution of even and odd n in the input file(s). After sieving to
      Message 2 of 2 , Aug 21, 2013
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        --- In primenumbers@yahoogroups.com, "robert44444uk" <robert_smith44@...> wrote:
        >
        > Recently I announced the discovery of a super-prime power series of the form k*2^n-1, n variable, k fixed, with over 200 primes to the Yahoo Primeform group.
        >
        > http://tech.groups.yahoo.com/group/primeform/message/11407
        >
        > An analysis of the primes discovered to date shows that there are significantly more even n in the list of primes (119) than odd (94). The complete list of the 213 primes to date is at:
        >
        > http://www.mersenneforum.org/showthread.php?t=18407 message 5.


        The occurence of more even exponents in the list of primes is a direct result of the distrubution of even and odd n in the input file(s). After sieving to p=6.34B the input file for the range n=500,001-800,000 contains 30872 even and 26912 odd n candidates.
        That's a ratio of 53.4% : 46.6%.
        The ratio for the 213 primes is quite similar: 55.9% : 44.1%.

        The occurence of a specific n in a sieve file for a given k (and therefore even and/or odd n) is determined by the residue classes (k mod p) and (b^n mod p) and the relations between them (e.g. for eliminating a specific n in k*b^n-1 we need to have (k mod p) == 1/(b^n mod p) ).

        For simpler (smaller) k there might be cases with no even or odd exponents at all, for example: 5*2^n-1 (no odd exponents) or 7*2^n-1 (no even ones).

        I would therefore expect also some Payam sequences with an opposite distribution, e.g. more odd than even exponents.

        -- Thomas
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