A113274 Record gaps between twin primes.
2, 6, 12, 18, 30, 36, 72, 150, 168, 210, 282, 372, 498, 630, 924, 930, 1008, 1452, 1512, 1530, 1722, 1902, 2190, 2256, 2832, 2868, 3012, 3102, 3180, 3480, 3804, 4770, 5292, 6030, 6282, 6474, 6552, 6648, 7050, 7980, 8040, 8994, 9312, 9318, 10200, 10338, 10668
Offset: 1
Keywords
Examples
The first twin primes are 3,5 and 5,7 so a(0)=5-3=2. The following pair is 11,13 so a(1)=11-5=6. The following pair is 17,19 so 6 remains the record and no terms are added.
Links
- Martin Raab, Table of n, a(n) for n = 1..82 (terms up to a(72) from Alexei Kourbatov, a(73)-a(75) from Tomás Oliveira e Silva)
- Richard Fischer, Maximale Lücken (Intervallen) von Primzahlenzwillingen
- G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
- Alexei Kourbatov, Maximal gaps between prime k-tuples
- Alexei Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013; and J. Int. Seq. 16 (2013) #13.5.2
- Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053[math.NT], 2013.
- Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
- Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
- Mersenneforum, Gaps between prime pairs (Twin Primes).
- Tomás Oliveira e Silva, Gaps between twin primes
- Luis Rodriguez and Carlos Rivera, Conjecture 66. Gaps between consecutive twin pairs, The Prime Puzzles and Problems Connection.
- Eric Weisstein's World of Mathematics, k-Tuple Conjecture
- Eric Weisstein's World of Mathematics, Twin Prime Constant
Crossrefs
Programs
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Mathematica
NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k+=6]; k]; p = 5; r = 2; t = {2}; Do[ q = NextLowerTwinPrim[p]; If[q > r + p, AppendTo[t, q - p]; Print[{p, q - p}]; r = q - p]; p = q, {n, 10^9}]; t (* Robert G. Wilson v, Oct 22 2005 *) DeleteDuplicates[Differences[Select[Partition[Prime[Range[10^7]],2,1],#[[2]]-#[[1]] == 2&][[All,2]]],GreaterEqual] (* The program generates the first 27 terms of the sequence. *) (* Harvey P. Dale, Dec 31 2022 *)
Formula
a(n) = A036063(n) + 2.
From Alexei Kourbatov, Dec 29 2011: (Start)
(1) Upper bound: gaps between twin primes are smaller than 0.76*(log p)^3, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a(log(p/a)-1.2), where a = 0.76*(log p)^2 is the average gap between twin primes near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.76 is reciprocal to the twin prime constant 1.32032...
(End)
Extensions
More terms from Robert G. Wilson v, Oct 22 2005
Corrected terms based on A036063, cross-checked with independent computations by Carlos Rivera and Richard Fischer (linked).
Terms up to a(72) are given in Kourbatov (2013), terms up to a(75) in Oliveira e Silva website.
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