## Measuring prime gaps

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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
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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