New evidence for the infinitude of some prime constellations

Thomas R. Nicely

                    http://www.trnicely.net

ABSTRACT

A conjecture of Hardy and Littlewood, implying the twin-primes conjecture and featured in the work of Arenstorf, is extended to some other prime constellations. Computational evidence is presented in support of the conjectures, which imply the infinitude of these constellations, including the twin primes.

MATHEMATICS SUBJECT CLASSIFICATION (MSC2000)

Primary: 11A41. Secondary: 11N05, 11-04.

KEYWORDS

Prime constellations, twin-primes conjecture, prime triplets, prime quadruplets, prime ktuples conjecture, Hardy-Littlewood approximations.

EXPOSITION

One of the famous unsettled questions in number theory is the “twin-primes conjecture”: there exists an infinite number of primes p for which p + 2 is also prime. R. F. Arenstorf [1] recently set forth a proposed proof of the twin-primes conjecture, employing analytic continuation, the complex Tauberian theorem of Wiener and Ikehara, and other advanced techniques of classical analytic number theory. Following the discovery of a significant error [8], the paper was withdrawn for revision. However, the genesis of the present paper is in Arenstorf’s strategy, which was based on an attempt to prove a related stronger conjecture of Hardy and Littlewood [4],

(1) lim_{n→∞} F_{2}(n) = 2c_{2} = 1.32032363169373914785562422002911155686524672 …,

where

(2) F_{2}(n) = \frac{1}{n} ∑_{p≤n} ln p · ln (p + 2) ,

the sum being taken over all twin-prime pairs (p, p + 2) with p ≤ n (Arenstorf uses p < n, but the difference is trivial, and the convention here is the one adopted in several previous studies of the twin primes). The constant c_{2} is the twin-primes constant, defined as

(3) c_{2} = ∏_{p>2} \frac{p(p − 2)}{(p − 1)^{2}} = ∏_{p>2} [1 − \frac{1}{(p − 1)^{2}} ] = 0.66016181584686957392781211 …,

the infinite products being taken over all primes except 2; c_{2} was computed to 42 decimals by Wrench [10], and eventually to more than 1000 decimals by Moree and Niklasch [5]. Note that some authors refer to 2c_{2} as the twin-primes constant.

Proving that the limit in (1) is 2c_{2}, or indeed any positive constant, is tantamount to proving the twin-primes conjecture; for if the twin primes were finite in number, the sum in (2) would necessarily be bounded, and the limit would be zero. This is in contrast to the sum of the reciprocals of the twin primes, which was proven [2] to converge to “Brun’s constant”, B_{2} = 1.90216 05826 22 ± 0.00000 00016 04 [6], whether or not the twin primes are infinite in number.

In support of (1), Wolf [9] computed F_{2}(n) for n = 2^{k} up to 2^{40} ≈ 1.1 × 10^{12}, and observed excellent convergence toward the predicted limit. Additional such numerical evidence is presented as part of this paper.

The twin-primes constellation consists of all prime pairs of the form (p, p + 2), beginning with (3, 5) and (5, 7). We will consider three additional prime constellations—two sets of prime triplets and a set of prime quadruplets. These are chosen for several reasons. The author has been studying them for some time in other contexts; they are in some sense among the simplest constellations other than the twins; and historically they were among the first constellations to be studied [3, 7]. We thus consider the set Q_{3a} of all prime triplets of the form (p, p + 2, p + 6), beginning with (5, 7, 11); the set Q_{3b} of all prime triplets of the form (p, p + 4, p + 6), beginning with (7, 11, 13); and the set Q_{4} of all prime quadruplets of the form (p, p + 2, p + 6, p + 8), beginning with (5, 7, 11, 13). The corresponding counting functions, indicating the number of members of each set such that p ≤ n, are denoted by π_{3a}(n), π_{3b}(n), and π_{4}(n).

Each of these three prime constellations, like the twins, is admissible; that is, the members do not form a complete residue class with respect to any prime. In consequence, the prime k-tuples conjecture of Hardy and Littlewood [4] implies the infinitude of each of these constellations. A classic example of an inadmissible constellation is the set of prime triplets of the form (p, p+2, p+4), which constitute a complete residue class modulo 3, and therefore must be finite in number (the only instance being (3, 5, 7)).

Hardy and Littlewood’s prime k-tuples conjecture [4] furthermore implies specific (asymptotic) densities for admissible prime constellations, with consequent integral formulas (the Hardy-Littlewood approximations) approximating the counts of each such constellation; these are analogues of the Prime Number Theorem, π(x) ∼ Li(x). Here we use the notation that f(x) is asymptotic to g(x) as x → ∞, f(x) ∼ g(x), if and only if lim_{x→∞} f(x)/g(x) = 1. In particular, the Hardy-Littlewood approximation for the count π_{2}(n) of twin primes is [4, p. 44]

(4) π_{2}(n) ∼ 2c_{2} ∫_{2}^{n} \frac{dt}{ln^{2} t} .

In fact, conjecture (1) can be deduced from conjecture (4), as follows. If dP represents the number of twin-prime pairs in an interval of measure dt,

(5) F_{2}(n) ∼ \frac{1}{n} ∫_{t=2}^{t=n} ln t · ln(t + 2) dP

∼ \frac{1}{n} ∫_{t=2}^{t=n} ln^{2} t dP

∼ \frac{1}{n} ∫_{2}^{n} ln^{2} t \frac{2c_{2} dt}{ln^{2} t} , by (4);

∼ \frac{1}{n} ∫_{2}^{n} 2c_{2} dt

F_{2}(n) ∼ \frac{1}{n} ∫_{0}^{n} 2c_{2} dt

∼ \frac{1}{n}2c_{2}n

∼ 2c_{2} ,

from which (1) follows.

In a similar fashion, we can start from the Hardy-Littlewood approximations [4] for other admissible prime constellations and deduce conjectured relationships analogous to (1). Relevant to the three additional prime constellations we have chosen to consider, we thus define

F_{3a}(n) = \frac{1}{n} ∑_{p≤n} ln p · ln (p + 2) · ln (p + 6) , (6a)

F_{3b}(n) = \frac{1}{n} ∑_{p≤n} ln p · ln (p + 4) · ln (p + 6) , (6b)

(6c) F_{4}(n) = \frac{1}{n} ∑_{p≤n} ln p · ln (p + 2) · ln (p + 6) · ln (p + 8) .

Here F_{3a} is evaluated over Q_{3a}, F_{3b} is evaluated over Q_{3b}, and F_{4} is evaluated over Q_{4}. The Hardy-Littlewood integral approximations for these three constellations are as follows [7, p. 61].

π_{3a}(n) ∼ \frac{9}{2}c_{3} ∫_{2}^{n} \frac{dt}{ln^{3} t} , (7a)

π_{3b}(n) ∼ \frac{9}{2}c_{3} ∫_{2}^{n} \frac{dt}{ln^{3} t} , (7b)

π_{4}(n) ∼ \frac{27}{2}c_{4} ∫_{2}^{n} \frac{dt}{ln^{4} t} . (7c)

The constants c_{3} and c_{4} are additional instances (along with c_{2}) of the Hardy-Littlewood constants [4], defined in general for k = 2, 3, 4, …, by an infinite product over primes,

(8) c_{k} = ∏_{p>k} \frac{p^{k−1}(p − k)}{(p − 1)^{k}} .

In particular,

c_{3} = ∏_{p>3} \frac{p^{2}(p − 3)}{(p − 1)^{3}} = 0.635166354604271207206696591272522417342 …, (9a)

c_{4} = ∏_{p>4} \frac{p^{3}(p − 4)}{(p − 1)^{4}} = 0.307494878758327093123354486071076853 …, (9b)

where the decimal approximations are taken from Moree and Niklasch [5], who have computed the Hardy-Littlewood and associated constants to 1002 decimals.

Derivations parallel to that of (5) now lead to the following new conjectures.

(10a) lim_{n→∞} F_{3a}(n) = \frac{9}{2}c_{3} = 2.858248595719220432430134660726350878 …,

(10b) lim_{n→∞} F_{3b}(n) = \frac{9}{2}c_{3} = 2.858248595719220432430134660726350878 …,

(10c) lim_{n→∞} F_{4}(n) = \frac{27}{2}c_{4} = 4.15118086323741575716528556195953751579941 ….

We remark that the aggregate constants appearing in (10) are denoted as D (in (10a) and (10b)) and E (in (10c) by Moree and Niklasch [5].

Note that, as with (1) and the twin-primes conjecture, conjectures (10) imply the infinitude of the associated prime constellations; indeed, the existence of any positive values for the limits would be sufficient, since the existence of only a finite number of a constellation would bound the corresponding sum in (6) and produce a limit of zero. Furthermore, any one of conjectures (10) would imply the twin-primes conjecture as well, since each triplet or quadruplet includes a twin-prime pair. In addition, conjectures (1) and (10) would also provide strong evidence in favor of the Hardy-Littlewood approximations and the prime k-tuples conjecture, of which they are consequences.

In support of conjectures (1) and (10), C code already designed for investigation of the prime constellations was modified to compute and record values of F_{2}(n), F_{3a}(n), F_{3b}(n), and F_{4}(n). The code was executed on personal computers using the Wintel environment and floating point arithmetic to long double precision (64-bit mantissa). Results to 10^{11} were checked by means of a second code, using another (slower) algorithm. Partial results are shown in Table 1, for n = 10^{k}, k = 1, …, 14. The last line indicates the conjectured limiting values.

Table 1. \begin{matrix}n & F_{2}(n) & F_{3a}(n) & F_{3b}(n) & F_{4}(n) \\ 10^{1} & 0.489996983652836 & 0.750978013334680 & 1.196828133529755 & 1.926220572770529 \\ 10^{2} & 0.799745092944858 & 1.048692856826072 & 2.585524317018651 & 0.705708858983092 \\ 10^{3} & 1.028368743410726 & 2.431181265229706 & 2.452935721305665 & 3.346148475381936 \\ 10^{4} & 1.295058615919660 & 2.506112075929775 & 2.606434053615298 & 3.410982520765379 \\ 10^{5} & 1.302244190054408 & 2.766307011496849 & 2.596941911928010 & 3.696922752013137 \\ 10^{6} & 1.309190742738575 & 2.778357571785003 & 2.879280996080815 & 4.083968787517717 \\ 10^{7} & 1.326032793752577 & 2.853365876050076 & 2.889851473019375 & 4.405342771818259 \\ 10^{8} & 1.320064585311069 & 2.868713724586362 & 2.860412298570633 & 4.172717820117937 \\ 10^{9} & 1.320027197802376 & 2.856981682913185 & 2.857859746634257 & 4.151690282083630 \\ 10^{10} & 1.320400512636363 & 2.856403308005203 & 2.854899772709739 & 4.137502967905835 \\ 10^{11} & 1.320369198427433 & 2.858844688347779 & 2.857182234184146 & 4.149169959064297 \\ 10^{12} & 1.320341275524246 & 2.858611546644537 & 2.858473729549299 & 4.153135542402516 \\ 10^{13} & 1.320329462705253 & 2.858318952277267 & 2.858405338766327 & 4.150776515382274 \\ 10^{14} & 1.320324077892631 & 2.858246127485545 & 2.858260937723365 & 4.151257032735108 \\ +∞ & 1.320323631693739 & 2.858248595719220 & 2.858248595719220 & 4.151180863237416 \end{matrix}

The convergence of F_{2}(n), F_{3a}(n), F_{3b}(n), and F_{4}(n) toward the conjectured limits appears to be quite convincing; the values for F_{2}(n) complement and are consistent with those of Wolf [9]. Of course, this does not constitute a proof of conjectures (1) and (10), but it does stand as persuasive evidence for them, and consequently for the infinitude of the corresponding prime constellations, as well as for the associated Hardy-Littlewood approximations and the prime k-tuples conjecture.

In regard to F_{2}(n) and the definitive proof of the twin-primes conjecture, we await remediation of the deficiencies discovered in Arenstorf’s work [1]. If this is accomplished, it may be possible to extend Arenstorf’s technique to the other three constellations, and perhaps even to any admissible prime constellation, proceeding from straightforward modifications of (6) and (10).

ACKNOWLEDGMENT

My thanks to Professor Richard P. Brent of the Oxford University Computing Laboratory for directing me to Arenstorf’s paper.

References

1. R. F. Arenstorf, There are infinitely many twin primes, arXiv:math.NT/0405509 v1, electronic (26 May

2004), posted at ≪http://arxiv.org/abs/math.NT/0405509≫. Withdrawn 8 June 2004.

2. Viggo Brun, La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+... où les dénominateurs sont “nombres premieres jumeaux” est convergente ou finie, Bulletin des Sciences Mathématiques 43 (1919), 100–104, 124–128.

3. F. J. Gruenberger and G. Armerding, Statistics on the first six million primes, UMT 73, Math. Comp. 19 (1965), 503–505.

4. 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 Math. 44 (1922/1923), 1–70. See also Collected papers of G. H. Hardy, Clarendon Press, Oxford, 1966, Vol. I, 561–630.

5. Pieter Moree and Gerhard Niklasch, Ultraprecision number-theoretical constants, electronic (23 August

2002), posted at ≪http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml≫.

6. Thomas R. Nicely, Prime constellations research project, electronic (18 August 2004), posted at

≪http://www.trnicely.net/counts.html≫.

7. Hans Riesel, Prime numbers and computer methods for factorization, 2nd ed., Birkhäuser, Boston, 1994, pp. 60–83. ISBN 0-8176-3743-5. MR 95h:11142.

8. Gerald Tenenbaum, Re: Arenstorf’s paper on the twin prime conjecture, Number Theory Mailing List ≪NMBRTHRY@LISTSERV.NODAK.EDU≫, electronic (8 June 2004), posted at

≪http://listserv.nodak.edu/scripts/wa. exe?A2=ind0406&L=nmbrthry&F=&S=&P=1119≫.

9. Marek Wolf, Numerical evidence in favor of the Arenstorf formula, arXiv:math.NT/0406091 v1, elec-

tronic (5 June 2004), posted at ≪http://arxiv.org/abs/math.NT/0406091≫.

10. John W. Wrench, Jr, Evaluation of Artin’s constant and the twin-prime constant, Math. Comp. 15 (1961), 396–398. MR 23:A1619.

DOCUMENT HISTORY

22 Oct 2004 Last modified. 20 July 2004 First posting. 15 July 2004 Rejected by Virginia Journal of Science. 12 July 2004 Revised version submitted to Virginia Journal of Science. 21 June 2004 Rejected by Journal of Integer Sequences. 19 June 2004 Submitted to Journal of Integer Sequences.

Aside from minor revisions and corrections, the content of this document is essentially that of the first posting.

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