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Goldbach conjecture verificationGoldbach conjecture verification Introduction News Results Top 20 Contributors References Links Contact [Up] IntroductionThe Goldbach conjecture is one of the oldest unsolved problems in numbertheory [1, problem C1]. In its modern form, it states thatevery even number larger than two can be expressed as a sum of two prime numbers.Let n be an even number larger than two, and let n=p+q, with p andq prime numbers, p<=q, be a Goldbach partition of n.Let r(n) be the number of Goldbach partitions of n. The number ofways of writing n as a sum of two prime numbers, when the order of the two primes isimportant, is thus R(n)=2r(n) when n/2 is not a prime and is R(n)=2r(n)-1when n/2 is a prime. The Goldbach conjecture states that r(n)>0, or,equivalently, that R(n)>0, for every even n larger than two.In their famous memoir [2, conjecture A], Hardy and Littlewoodconjectured that when n tends to infinity, R(n) tends asymptotically to (i.e.,the ratio of the two functions tends to one) n p-1N2(n) = 2 C ---------------- PRODUCT --- , twin (log n)(log n-2) p odd prime p-2 divisor of nwhere p(p-2)C = PRODUCT ------- = 0.66016181584686957392... twin p odd prime (p-1)^2is the twin primes constant. In [3], Crandall and Pomerance suggestreplacing the factor n----------------(log n)(log n-2)appearing in the formula of N2(n) by the asymptotically equivalent factor n-2 dxINTEGRAL --------------- . 2 log(x) log(n-x)The numerical evidence supporting this conjectured asymptotic formula is very strong. Up to10^10, the Crandall-Pomerance formula does not deviate from R(n) by more than40150.Let us order the r(n) Goldbach partitions of n by increasing order of thesmallest prime of the partition. More precisely, let us denote the two primes of the i-thGoldbach partition of n by p(n;i) and q(n;i), withp(n;i) <= q(n;i) and p(n;i) < p(n;i+1).In order to verify the Goldbach conjecture for a given n, it is sufficient to find one ofits Goldbach partitions. Our strategy will be to find the minimal Goldbach partitionn=p(n;1)+q(n;1), i.e., the one that uses the smallest possible primenumber p(n)=p(n;1). As in [4], for every prime qwe will denote by S(q) the least even number n suchthat p(n)=q.NewsFebruary 1, 2005: 2·10^17 reached.March 20, 2005: 10^17 double checked.December 26, 2005: 3·10^17 reached.June 5, 2006: 4·10^17 reached.September 28, 2006: interval between 7·10^17 and 10^18 checked.February 19, 2007: 5·10^17 reached.February 23, 2007: interval between 6·10^17 and 7·10^17 checked.April 25, 2007: 10^18 reached.February 16, 2008: 11·10^17 reached.July 14, 2008: 12·10^17 reached.Computational resultsWe have implemented a program that finds the minimal Goldbach partition of every even integerlarger than four. In order to do this efficiently, the computation intensive parts of theprogram were written in assembly language (for the IA32 instruction set). A very efficientcache friendly implementation of the segmented sieve ofEratosthenes was used to generate the prime numbers (see ourspeed comparison chart [17k, PDF]between several Intel and AMD CPUs). For each interval of 10^12 integers, we record thenumber of times each (small) prime is used in a minimal Goldbach partition, as well as the eveninteger where it was first needed. Because it takes very little extra time, we also recordinformation about the gaps between consecutive primes, viz., how manytimes each gap occurs, and its first occurrence. On a 2.2GHz Athlon64 3500+ processor, testing aninterval of 10^12 integers near 10^18 takes close to 75 minutes. The executiontime of the program grows very slowly, like log(N), where N is the last integer ofthe interval being tested, and it uses an amount of memory that is roughly given by13 sqrt(N) / log(N). The program is now running on the spare time of around 50computers (20 at DETI/UA and 30 at PSU), either under GNU/Linux or under Windows2000/XP. We have reached 10^18 in April 2007, and are now extending ourcomputations to 1.5·10^18.The following table presents an overview of the current status of thismassive computation. Each cell represents an interval of 10^15; its background colorindicates its computational status (green for double-checked, yellow for single-checked, and redfor not yet done or not yet fully checked), and its brightness indicates if prime countsare available to perform an initial check of the correctness of the results (bright) or not(not so bright).00000001000200030004000500060007000800090010001100120013001400150016001700180019... (same state) ...0080008100820083008400850086008700880089009000910092009300940095009600970098009901000101010201030104010501060107010801090110011101120113011401150116011701180119012001210122012301240125012601270128012901300131013201330134013501360137013801390140014101420143014401450146014701480149015001510152015301540155015601570158015901600161016201630164016501660167016801690170017101720173017401750176017701780179... (same state) ...032003210322032303240325032603270328032903300331033203330334033503360337033803390340034103420343034403450346034703480349035003510352035303540355035603570358035903600361036203630364036503660367036803690370037103720373037403750376037703780379... (same state) ...096009610962096309640965096609670968096909700971097209730974097509760977097809790980098109820983098409850986098709880989099009910992099309940995099609970998099910001001100210031004100510061007100810091010101110121013101410151016101710181019... (same state) ...126012611262126312641265126612671268126912701271127212731274127512761277127812791280128112821283128412851286128712881289129012911292129312941295129612971298129913001301130213031304130513061307130813091310131113121313131413151316131713181319... (same state) ...1420142114221423142414251426142714281429143014311432143314341435143614371438143914401441144214431444144514461447144814491450145114521453145414551456145714581459146014611462146314641465146614671468146914701471147214731474147514761477147814791480148114821483148414851486148714881489149014911492149314941495149614971498149915001501150215031504150515061507150815091510151115121513151415151516151715181519... (same state) ...19601961196219631964196519661967196819691970197119721973197419751976197719781979198019811982198319841985198619871988198919901991199219931994199519961997199819992000200120022003200420052006200720082009201020112012201320142015201620172018201920202021202220232024202520262027202820292030203120322033203420352036203720382039... (same state) ...29602961296229632964296529662967296829692970297129722973297429752976297729782979298029812982298329842985298629872988298929902991299229932994299529962997299829993000300130023003300430053006300730083009301030113012301330143015301630173018301930203021302230233024302530263027302830293030303130323033303430353036303730383039... (same state) ...3960396139623963396439653966396739683969397039713972397339743975397639773978397939803981398239833984398539863987398839893990399139923993399439953996399739983999So far, we have tested all consecutive even numbers up to 1298·10^15, and double-checked our results up to 10^17.Some (87) extra intervals of 10^15 were also tested. About 382.7 CPU years were used to do all this. At least 34.62% of the work necessary to reach 4·10^18 is already done.As far as we are aware, the previous record of computation was4·10^14 [5]. As expected, no counter-example of the conjecturewas found. In this table [23k, compressed with gzip]we present all values of S(p) we were able to compute, as well as counts of the number oftimes each (small) prime was used in a minimal Goldbach partition. The record-holders, i.e.,numbers larger than all previous ones of the same kind, are clearly marked in the table, whichextends the tables 1 and 2 of [4] and table 1of [5]. The following figure presents a graph with the available valuesof S(p); the cyan dots represent data which is not known for certain to be a first occurrence. The values of S(p) are bounded, for our empirical data, by the functions 0.4 0.4 0.4 p 0.4 pS_min(p) = 0.06 p e and S_max(p) = 11.05 p e .The two kinds of record-holders marked in the table mentioned above correspond to the values ofS(p) closer to one of the two bounds. For large p the values of S(p) appearto be slowly approaching the upper bound; hence, the asymptotic growth rate of S(p)probably has a different functional form. In [6] it was stated, based onprobabilistic considerations, that p should not grow faster thanlog^2 S(p) log log S(p). This appears to be false (for this to be true the data pointsshould be above the black line of the figure, which is clearly not the case). Indeed, it appearsthat the growth rate of the black curve is too large. Thus, the estimatelog^2 S(p) log log S(p) appears to be too small. It was found that for all our datap can be reasonably well approximated by 0.33(log S(p) log log S(p))^2.Let D(x;p) be the relative frequency of occurrence of the prime p in theminimal Goldbach partition of the even numbers not larger than x. The followingfigure presents a graph of this function, computed for our current verification limit of theGoldbach conjecture. Besides the expected near exponential decay of D(x;p), it is interesting to observe thatthere exists a distinct difference of behavior in the values of this function when p is amultiple of three plus one (white dots) and when it is not (yellow dots).Cyan dots represent primes of minimal Goldbach partitions known to occur before 4·10^18,but which are outside of the interval used to make this graph.Top 20The following table presents the 20 largest p (least primes of a Goldbach partition) foundso far, with S(p)<4·10^18, either by contributors to this project or by other discoverers(those have an * before the discoverer name). Repeated values of p were excluded from this list. Aquestion mark indicates a first known occurrence, that is, the true value of S(p) may be smallerthan the one given. p S(p) Discoverer 9341 906 03057 95622 79642 John Fettig & Nahil Sobh 9161 887 12380 30778 37868 Siegfried "Zig" Herzog 8951 914 47723 42519 16254 John Fettig & Nahil Sobh 8941 555 27435 15567 50822 Siegfried "Zig" Herzog 8933 258 54942 69161 49682 Siegfried "Zig" Herzog 8737 764 63115 78502 42766 Siegfried "Zig" Herzog 8699 ? 2994 28857 66127 17268 Tomás Oliveira e Silva 8681 771 06523 23704 26528 Siegfried "Zig" Herzog 8663 1262 36426 84331 28726 Siegfried "Zig" Herzog 8641 517 71184 25980 37624 Tomás Oliveira e Silva 8629 1238 28931 11321 16112 Siegfried "Zig" Herzog 8623 1211 65003 16991 77606 Tomás Oliveira e Silva 8573 1134 05983 29344 00206 Siegfried "Zig" Herzog 8563 280 46026 69116 44252 Tomás Oliveira e Silva 8539 941 90839 15563 21548 Siegfried "Zig" Herzog 8527 1295 15748 05397 26954 Siegfried "Zig" Herzog 8521 1176 80059 43587 37918 Siegfried "Zig" Herzog 8501 255 32912 66885 55994 Siegfried "Zig" Herzog 8443 121 00502 23040 07026 Tomás Oliveira e Silva 8389 935 18588 11050 28832 Siegfried "Zig" Herzog ContributorsThe following table presents some details about the contribution of all (past and present)individuals which donated computing power to this project. Name Location Number of tasks Number of first (known) occurrences Minimal Goldbach partitions Prime gaps Siegfried "Zig" Herzog PSU 814509 68 (0) 45 (0) Tomás Oliveira e Silva All 658496 297 (5) 51 (3) DETUA 573525 173 (4) 40 (2) IEETA 30492 121 (0) 6 (0) Home 54479 3 (1) 5 (1) John Fettig & Nahil Sobh NCSA 33641 2 (0) 1 (0) António Teixeira IEETA 14995 0 (0) 0 (2) João Manuel Rodrigues DETUA 10238 9 (0) 1 (1) Christian Kern Germany 9000 0 (2) 0 (0) Carlos Bastos DETUA 8646 11 (0) 1 (0) Rui Arnaldo Costa IEETA 3847 4 (0) 0 (0) Armando Pinho IEETA 3285 2 (0) 1 (0) Miguel Oliveira e Silva DETUA 2659 7 (0) 0 (0) Laurent Desnoguès France 200 0 (0) 0 (0) All All 1559516 400 (7) 100 (6) This work was partially supported by the National Center for Supercomputing Applicationsand utilized the NCSA Xeon cluster.References[1]Richard K. Guy,Unsolved problems in number theory,third edition, Springer-Verlag, 2004.[2]G. H. Hardy andJ. 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.[3]Richard Crandall andCarl Pomerance,Prime numbers: a computational perspective,Springer-Verlag, 2001.[4]Matti K. Sinisalo,Checking the Goldbach conjecture up to 4·10^11,Mathematics of Computation, vol. 61, no. 204, pp. 931-934, October 1993.[5]Jörg Richstein,Verifying the Goldbach conjecture up to 4·10^14,Mathematics of Computation, vol. 70, no. 236, pp. 1745-1749, July 2000.[6]A. Granville,J. van de Lune, andH. J. J. te Riele,Checking the Goldbach conjecture on a vector computer,in Number Theory and Applications, R. A. Mollin (ed.), pp. 423-433, Kluwer Academic Press, 1989.Additional linksOur prime gaps page.MathSoft's Hardy-Littlewood Constants page.Tomás Oliveira e SilvaDepartamento de Electrónica, Telecomunicações e InformáticaUniversidade de Aveiro3810-193 AVEIROPORTUGAL December 16, 2008 Phone: +351-234-370379Fax: +351-234-370545 Phone (internal): 23013Office: DET 237E-mail address: tos@ua.ptHome page: http://www.ieeta.pt/~tos/ |
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