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Concerning the largest gap between certain number pairs and some working on that

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  • chrisdarroch
    Hi, Would those who will.........consider the following series of number pairs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 .........210 2 3 4 5 6 7 8 9
    Message 1 of 1 , Jul 29, 2004
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      Hi,

      Would those who will.........consider the following series of number
      pairs.

      1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 .........210
      2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .........211

      take out any pair which includes any number divisible by 3...leaving

      1 4 7 10 13 16 19...............208
      2 5 8 11 14 17 20...............209

      Take out all of the pairs divisible by 5.....leaving

      1 7 13 16 ......................208
      2 8 11 17 ......................209

      Do the same for 7.

      The number of gaps formed will be 30, and this can be found by the
      Chinese Remainder Theroem (I believe).

      I am looking for a formula which will tell me about the largest gap
      that will be left between two consecutive pairs.

      Obviously for this one example I can use observation and say that
      there is at least one gap of size 15, which is the largest.

      But I am also interested in the largest gap for the series of pairs
      ending in 2.3.5.7.11 (2310) pairs; and of course the series of pairs
      ending in 2.3.5.7.11.13 and so on.

      I need to place some kind of bound on the largest gap that could
      exist between two consective pairs in such series.

      Let me return to my first example to show some of my thinking on the
      subject.

      By the CRT I can see that out of 210 pairs, there remain 30 gaps.

      I may approach the subject of the largest one by segregating out one
      gap as "that largest" and the other 29 as equivalent or smaller.

      I can say that due to the extraction of all pairs containing a number
      divisible by 3 the gap between any consecutive pair remaining must be
      3 and thus I have 29 gaps which are 3 at least.

      Up till then I can say that 3.29 gaps cover an area of 87 and thus
      the other gap must be at most 210 - 87 = 123

      If I consider the fact that more of the pairs are extracted due to
      containing numbers divisible by 5 then, I can say after that
      extraction, some gaps will be 3 and others form gaps of 6.

      I can say that 210 /(3.5) gaps will be 6 i.e. 14 gaps will be of size
      6.

      Thus I can say that out of the 29 gaps 14 of them must cover an area
      of at least 6 and the remaining 15 can be 3.

      Thus I can say that so far the 29 cover an area of

      (6.14) + (3.15) = 129

      Thus the remaining single gap must at this point be at most

      210 - 129 = 81

      The difficulty begins when I have to consider other factors.
      Can anyone see this line of reasoning being sensible?

      Are there alternatives.

      For example can one use something of the results for the largest gap
      between primes such as:

      ยท g(pn) < (1/16597)pn for n > 2010760 (Schoenfeld 1976)

      to form an estimate for the largest gap in my series?

      Chris
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