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De Polignac's conjecture - an observation

  1. Nov 2, 2012 #1
    I have been playing around with an idea related to de Polignac's conjecture - I was hoping someone in the group could guide me further:

    Given the nth prime p(n), in the interval [p(n)+1 .. p(n)^2-1] the prime differences are the same as the difference sequence for the reduced residue system of the primorial p(n)#. So for example, in the interval [12 .. 120] the prime differences are given by OEIS A049296 (and the primes themselves by OEIS A008364).

    The commentary on OEIS A049296 notes that these sequences of differences of reduced residue systems of the primorials are periodic and near-palindromic (with an extra term of 2). A given sequence (say, of differences of the RRS of p(n)#) is obtained from the preceding one (relating to p(n-1)#) by combining those terms that correspond to the higher multiples of p(n). (This is equivalent to removing the multiples of p(n) in a sieve of Eratosthenes)

    It occurred to me that - for a given large even number E, there may be no sub-sequence of differences of the RRS of a large p(n)# that will sum to E. The same would then be true of the sequences relating to all higher primorials p(i>n)#, and (by the first paragraph above) E would be a counter-example to de Polignac's conjecture.

    Conversely, if it could be shown that every even number can be expressed as the sum of a sub-sequence of the differences of the reduced residue set of an arbitrarily large p(n)# it might be a step on the way to proving the conjecture.

    I took this to somebody at my school in a slightly more long-winded form a while ago, but the response was a bit vague and dismissive. I have tried to find relevant material in text books and on the net but can't find anything obviously related. So I would be very grateful for any comments from the group as to whether this is of interest, or pointers to related reading.
  2. jcsd
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