Wikipedia / Primes in arithmetic progression
Wikipedia / Primes in arithmetic progression
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression.
According to the Green–Tao theorem, there exist arbitrarily long, where a and b are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites.
For any integer, an AP-k (also called PAP-k. An AP-k to be the first prime in the arithmetic progression.
Properties
Any given arithmetic progression of primes has a finite length. In 2004, Ben J. Green and Terence Tao settled an old conjecture by proving the Green–Tao theorem: the primes contain arbitrarily long arithmetic progressions..
, then the common difference is a multiple of the primorial.
Proof:, then modular arithmetic.
.
. , for this AP-19 found by Wojciech Iżykowski in 2013:
.
It follows from widely believed conjectures, such as Dickson's conjecture and some variants of the prime k-tuple conjecture. For example, 5 is the smallest prime not dividing 6, so there is expected to be infinitely many AP-4 with common difference 6, which is called a sexy prime, it is the twin prime conjecture.
Minimal primes in AP
We minimize the last term.
Largest known primes in AP
denotes the primorial.
is an AP-27. Several examples are known for AP-26. The first to be discovered was found on April 12, 2010, by Benoît Perichon on a PlayStation 3 with software by Jarosław Wróblewski and Geoff Reynolds, ported to the PlayStation 3 by Bryan Little, in a distributed PrimeGrid project:
43142746595714191 + 23681770·23#·n, for n = 0 to 25. (23# = 223092870)
By the time the first AP-26 was found the search was divided into 131,436,182 segments by PrimeGrid and processed by 32/64bit CPUs, Nvidia CUDA GPUs, and Cell microprocessors around the world.
Before that, the record was an AP-25 found by Raanan Chermoni and Jarosław Wróblewski on May 17, 2008:
)
The AP-25 search was divided into segments taking about 3 minutes on Athlon 64 and Wróblewski reported "I think Raanan went through less than 10,000,000 such segments" (this would have taken about 57 cpu years on Athlon 64).
The earlier record was an AP-24 found by Jarosław Wróblewski alone on January 18, 2007:
.
For this Wróblewski reported he used a total of 75 computers: 15 64-bit Athlons, 15 dual core 64-bit Pentium D 805, 30 32-bit Athlons 2500, and 15 Durons 900.
with the year of discovery and the number of decimal.
Consecutive primes in arithmetic progression
Consecutive primes in arithmetic progression refers to at least three consecutive, all the other numbers between the terms of the progression must be composite. For example, the AP-3 {3, 7, 11} does not qualify, because 5 is also a prime.
, a CPAP-k. The middle prime in a CPAP-3 is called a balanced prime. The largest known has 15004 digits.
The first known CPAP-10 was found in 1998 by Manfred Toplic in the distributed computing project CP10 which was organized by Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann. This CPAP-10 has the smallest possible common difference, 7# = 210. The only other known CPAP-10 as of 2018 was found by the same people in 2008.
If a CPAP-11 exists then it must have a common difference which is a multiple of 11# = 2310. The difference between the first and last of the 11 primes would therefore be a multiple of 23100. The requirement for at least 23090 composite numbers between the 11 primes makes it appear extremely hard to find a CPAP-11. Dubner and Zimmermann estimate it would be at least 1012 times harder than a CPAP-10.
Minimal consecutive primes in AP
.
Largest known consecutive primes in AP
.xd is a d-digit number used in one of the above records to ensure a small factor in unusually many of the required composites between the primes. x106 = 115376 22283279672627497420 78637565852209646810 56709682233916942487 50925234318597647097 08315833909447378791 x153 = 9656383640115 03965472274037609810 69585305769447451085 87635040605371157826 98320398681243637298 57205796522034199218 09817841129732061363 55565433981118807417 = x253 modulo 379# x253 = 1617599298905 320471304802538356587398499979 836255156671030473751281181199 911312259550734373874520536148 519300924327947507674746679858 816780182478724431966587843672 408773388445788142740274329621 811879827349575247851843514012 399313201211101277175684636727
See also
Notes
References
Chris Caldwell, , and , all from the Prime Pages.
Jarosław Wróblewski,
P. Erdős and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.