where to find a prime gap and what is its minimum length
- i think you can find a gap anywhere the following way:
for a number = Pi X Pj X Pk X ... X Pz, where P is a prime number
and i, j, k, ... ,z are in sequence from lowest to hightest for that prime number, but they may be the same prime. e.g #1:
4290 = 2 X 3 X 5 X 11 X 13, the prime gap exists at 4290 - 2 (4288) and runs at least another 7 - 2 = 5 numbers to 4284, so the gap is 4288, 4287, 4286, 4285 & 4284. the minimum length is determined thusly:
arranging the prime factors in ascending order: take the prime number that is missing in the sequence of all prime numbers arranged in ascending order, (in this case 7) and subtract 2 to get a minimum gap length of 5.
12012 = 2 X 2 X 3 X 7 X 11 X 13, so the gap starts at 12012 - 2 = 12010 and is a minimum length of 5 - 2 = 3. 5 is the missing prime in the ascending order of the prime factors of 12012, even though there are 2 2s.
to determine the exact length of the prime gap would require an ability to determine the prime factors of successively smaller and successively larger even numbers, and apply the same formula: e.g. if we new the prime factors of the successivelyreferencing the next successively larger even number 12014 and the successively smaller even number of 12010, then we could establish the minimum lengths of the gaps starting at 12014 and 12010 starting at 12014 - 2 = 12012 and 12010 - 2 = 12008, respectively, and thus should by iteration be able to determine the exact length of the prime gap starting at any point we like.
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