"Possible" versus real gaps - and merit of large adjacent gaps
- In a previous post I discussed how every prime up to 43 was used to factor numbers in the
gap of 33 composites between 1327 and 1361. Clearly, then, what we have is a gap as large
as is potentially possible ie. there is no way if we assume the factors combine so efficiently as
to use them all in factoring numbers not divisible by 2, 3, or 5 that we could have a larger
However, an examination of five-digit primes suggests that for numbers much larger than
1361 gaps much larger than actually occur are feasible.
For instance, the 99 numbers from 58790 to 58888 contain only one prime (58831). Here, of
the 50 primes (excluding 2, 3 and 5) less than the square root of 58888, 24 occur as factors
of the 24 numbers not divisible by 2, 3, or 5. More significantly, of the first 28 primes after 5,
only eight do not occur as factor between 58790 and 58888 - the smallest absent primes
being 41, 47, 61 and 67.
As I see it this case of two adjacent gaps of 42 and 58 shows it would take a prime gap of,
say, 100 among five-digit numbers or 250 among six-digit numbers to be equivalent in real
merit to that between 1327 and 1361. At least theoretically with efficiently arranged factors,
such gaps would not be "impossible" even though we know they actually do not happen.