Largest Known Primes in AP
For prime q, q# denotes the primorial 2·3·5·7·...·q.
As of April 2010, the longest known AP-k is an AP-26, found on April 12, 2010 by Benoãt Perichon on a PlayStation 3 with software by Jaroslaw Wroblewski 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) (sequence A204189 in OEIS)
Before that the record was an AP-25 found by Raanan Chermoni and Jaroslaw Wroblewski on May 17, 2008:
- 6171054912832631 + 366384·23#·n, for n = 0 to 24. (23# = 223092870)
The AP-25 search was divided into segments taking about 3 minutes on Athlon 64 and Wroblewski 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 Jaroslaw Wroblewski alone on January 18, 2007:
- 468395662504823 + 205619·23#·n, for n = 0 to 23.
For this Wroblewski 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.
The following table shows the largest known AP-k with the year of discovery and the number of decimal digits in the ending prime. Note that the largest known AP-k may be the end of an AP-(k+1). Some record setters choose to first compute a large set of primes of form c·p#+1 with fixed p, and then search for AP's among the values of c that produced a prime. This is reflected in the expression for some records. The expression can easily be rewritten as a·n + b.
| k | Primes for n = 0 to k−1 | Digits | Year | Discoverer |
|---|---|---|---|---|
| 3 | (11347·2508209 − 1) + (103939·2514229 − 11347·2508209)·n | 154804 | 2010 | David Broadhurst, Thomas Ritschel, Lei Zhou |
| 4 | (100997770 + 3624707n)·27751# + 1 | 11961 | 2008 | Ken Davis |
| 5 | (82751511 + 20333209n)·16229# + 1 | 7009 | 2009 | Ken Davis |
| 6 | (19303382 + 41724940n)·5011# + 1 | 2153 | 2009 | Ken Davis |
| 7 | (1246733996 + 35777939n)·3109# + 1 | 1328 | 2009 | Mike Oakes |
| 8 | (452558752 + 359463429n)·2459# + 1 | 1057 | 2009 | Ken Davis |
| 9 | (190556231 + 138880294n)·997# + 1 | 425 | 2009 | Ken Davis |
| 10 | (565429078 + 147743546n)·641# + 1 | 274 | 2009 | Mike Oakes |
| 11 | (197477410 + 146636n)·457# + 1 | 196 | 2009 | Jeff Anderson-Lee |
| 12 | (1366899295 + 54290654n)·401# + 1 | 173 | 2006 | Jeff Anderson-Lee |
| 13 | (1374042988 + 22886141n)·173# + 1 | 78 | 2006 | Mike Oakes |
| 14 | (145978014 + 253131151n)·157# + 1 | 71 | 2009 | Mike Oakes |
| 15 | (237375311 + 118560155n)·109# + 1 | 54 | 2009 | Mike Oakes |
| 16 | (281121075 + 18107251n)·83# + 1 | 42 | 2009 | Mike Oakes |
| 17 | (263013824 + 18107251n)·83# + 1 | 42 | 2009 | Mike Oakes |
| 18 | (1051673535 + 32196596n)·53# + 1 | 29 | 2007 | Jens Kruse Andersen |
| 19 | 62749659973280668140514103 + 107·61#·n | 27 | 2007 | Jaroslaw Wroblewski |
| 20 | 178284683588844176017 + 53#·n | 21 | 2007 | Jaroslaw Wroblewski |
| 21 | 28112131522731197609 + 19#·n | 20 | 2008 | Jaroslaw Wroblewski |
| 22 | 1351906725737537399 + 43#·n | 19 | 2008 | Jaroslaw Wroblewski |
| 23 | 117075039027693563 + 6548·23#·n | 19 | 2008 | Raanan Chermoni, Jaroslaw Wroblewski |
| 24 | 28806475189976381 + 36028618·23#·n | 18 | 2010 | John Petterson, PrimeGrid |
| 25 | 18626565939034793 + 30821486·23#·n | 18 | 2010 | Chris Wingate, PrimeGrid |
| 26 | 43142746595714191 + 23681770·23#·n | 18 | 2010 | Benoãt Perichon, PrimeGrid |
Read more about this topic: Primes In Arithmetic Progression
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