A235402 Mode of maximal "prime gaps" in Cramer's model with n urns.

1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Author

Alexei Kourbatov, Jan 10 2014

Keywords

• nonn

Comments

Cramer (1936) sets up his probabilistic model of primes as follows: "Let U1, U2, U3, ... be an infinite series of urns containing black and white balls, the chance of drawing a white ball from Un being 1/(log n) for n>2, while the composition of U1 and U2 may be arbitrarily chosen. We now assume that one ball is drawn from each urn, so that an infinite series of alternately black and white balls is obtained." White balls simulate "primes", and black balls are "composites".

Note that the model, as stated, is underdetermined: the content of urns U1 and U2 is arbitrary. To compute exact distributions of maximal gaps, here we assume that

U1 is empty — it produces neither "primes" nor "composites";

U2 always produces white balls (i.e., the number 2 is certain to be "prime").

It is not guaranteed that there are any "primes" > 2 at all. To circumvent this, we define the maximal prime gap as 1 + the longest run of "composites" observed in a given experiment with n urns.

Examples

			For n=3 we have only three urns: U1, U2, U3. Of these, only U3 produces random results:
- a white ball ("prime") with probability 1/(log3) ~ 0.91, or
- a black ball ("composite") with probability 1 - 1/(log 3).
Thus the longest run of "composites" is 0 with probability 0.91. Consequently, the maximal gap between "primes" is 1 with probability 0.91, so the mode (most probable value) of maximal "prime gap" is 1.
		

Links

• J. H. Cadwell, Large intervals between consecutive primes, Math. Comp. 25 (1971), No. 116, 909-913.
• H. Cramer, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 23-46.
• A. Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv:1401.6959 [math.NT], 2014.
• A. Kourbatov, Maximal gaps between Cramer's random primes from 2 to N: cdf, histogram, mode, median
• A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv preprint arXiv:1503.01744 [math.NT], 2015.
• Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht's Conjecture, arXiv preprint arXiv:1506.03042 [math.NT], 2015.
• A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2
• Alexei Kourbatov, On the distribution of maximal gaps between primes in residue classes, arXiv:1610.03340 [math.NT], 2016.
• Alexei Kourbatov, Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.

Crossrefs

Cf. A235492 (median of maximal "prime gaps" in Cramer's model).

Formula

a(n) = n log(li n)/(li n) + O(log n) = (log n)^2 - (log n)*(log log n) + O(log n), where li n is the logarithmic integral of n. The formula can be proved using results of Cramer (1936); Cadwell (1971) gives a derivation for the equivalent formula on the right (without li n).