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Measuring prime gaps

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  • julienbenney
    Over the last few hours, I have thought of measuring how large a prime gap is in terms of the ratio of the number of possible smallest prime factors to the
    Message 1 of 1 , Apr 12, 2004
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      Over the last few hours, I have thought of measuring how large a "prime
      gap" is in terms of the ratio of the number of possible smallest prime
      factors to the number of prime factors occurring within the gap.

      The most remarkable of all prime gaps, that between 1327 and 1361,
      contains every possible factor except 29 [(1361)^1/2 = 36.89173349]
      (which occurs in 1363 = 29*47) as follows [numbers divisible by 2, 3
      and 5 are excluded]:

      1331 = 11*11*11 1333 = 31*43 1337 = 7*191
      1339 = 13*103 1343 = 17*79 1349 = 19*71
      1351 = 7*193 1357 = 23*59

      The only two other gaps I have studied in this respect are:

      the exactly analogous gap between 8467 and 8501 - equalling the
      magnitude of the primal gap betwen 1327 and 1361, but with numbers
      6.3125 times larger:

      8471 = 43*197 8473 = 37*229 8477 = 7*7*173
      8479 = 61*139 8483 = 17*499 (obvious, really!)
      8489 = 13*653 8491 = 7*1213 8497 = 29*293

      Here, we see only seven of twenty-one prime factors lower than 8501^1/2
      (92.20086767) occurring as smallest prime factor - or one-third of the
      possible factors.

      and the prime gap between 31397 and 31469, which lasts until numbers
      five times as large. This is the second most persistent primal gap
      lower than one trillion.

      31399 = 17*1847 31403 = 31*1013 31409 = 7*7*641
      31411 = 101*311 31417 = 89*353 31421 = 13*2417
      31423 = 7*67*67 31427 = 11*2857 31429 = 53*593
      31433 = 17*43*43 31439 = 149*211 31441 = 23*1367
      31447 = 13*41*59 31451 = 7*4493 31453 = 71*443
      31457 = 83*379 31459 = 163*193 31463 = 73*431

      Given that 31469^1/2 = 177.3950394, there are thirty-seven possible
      prime factors less than the square root of the upper limit. Only
      eighteen actually divide any number between 31398 and 31468 not
      divisible by 2, 3, or 5 [31411 and 31427 can also be shown composite by
      elementary divisibility tests for 11 and 101]. This means that this is
      a "lesser" gap than that between 1327 and 1361 - for what size of gap
      could be theoretically made by using all the primes less than 177??
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