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Message boards : AP26 - AP27 Search : 23#

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DarrylProject donor
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Posts: 95
ID: 37210
Credit: 336,896,165
RAC: 222,625
Found 1 prime in the 2019 Tour de Primes321 LLR Gold: Earned 500,000 credits (503,464)Cullen LLR Gold: Earned 500,000 credits (500,149)ESP LLR Gold: Earned 500,000 credits (515,497)Generalized Cullen/Woodall LLR Gold: Earned 500,000 credits (527,019)PPS LLR Jade: Earned 10,000,000 credits (12,360,069)PSP LLR Amethyst: Earned 1,000,000 credits (1,006,254)SoB LLR Gold: Earned 500,000 credits (538,748)SR5 LLR Turquoise: Earned 5,000,000 credits (5,387,020)SGS LLR Sapphire: Earned 20,000,000 credits (20,006,208)TRP LLR Gold: Earned 500,000 credits (506,533)Woodall LLR Gold: Earned 500,000 credits (536,003)321 Sieve Jade: Earned 10,000,000 credits (10,011,269)Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (52,021,564)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,134,887)PPS Sieve Double Bronze: Earned 100,000,000 credits (100,260,654)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,761,921)TRP Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,020,224)AP 26/27 Double Bronze: Earned 100,000,000 credits (100,055,458)GFN Sapphire: Earned 20,000,000 credits (20,247,688)
Message 127380 - Posted: 24 Feb 2019 | 10:48:06 UTC

Hi All,

I just bought a new crunching rig with a decent graphics card, and decided to put it to work on the AP27 project. Looking at the first few results I found yesterday and today, and comparing with an ancient AP21 I found here at Primegrid 9 years ago (until I switched my new PC on yesterday this AP21 was the only AP I had ever found) I noticed that one common feature they all share is that the prime sequences are constructed by multiplying a given quantity by 23#. For example:

11988702023996197+4465007*23#*n for n=0..20

This made me curious as to whether there is any significance in the choice of 23# here (since it seems not to have changed in 9 years) - are any other sequences, constructed of say 29# or 17#, being tested as well? For numbers of the form a+b*c#*n for some range of n, it occurs to me we could get similar densities of primes and similar AP's as we are testing now, but using a different value than c=23.

Thanks
Darryl
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dukebg
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Found 1 prime in the 2018 Tour de Primes321 LLR Gold: Earned 500,000 credits (506,942)Cullen LLR Gold: Earned 500,000 credits (500,523)ESP LLR Gold: Earned 500,000 credits (655,642)Generalized Cullen/Woodall LLR Gold: Earned 500,000 credits (539,100)PPS LLR Gold: Earned 500,000 credits (688,232)PSP LLR Gold: Earned 500,000 credits (561,629)SoB LLR Ruby: Earned 2,000,000 credits (3,659,676)SR5 LLR Gold: Earned 500,000 credits (505,810)SGS LLR Gold: Earned 500,000 credits (506,024)TRP LLR Gold: Earned 500,000 credits (963,625)Woodall LLR Gold: Earned 500,000 credits (506,044)321 Sieve Gold: Earned 500,000 credits (511,694)Generalized Cullen/Woodall Sieve (suspended) Gold: Earned 500,000 credits (549,916)PPS Sieve Gold: Earned 500,000 credits (701,168)AP 26/27 Gold: Earned 500,000 credits (525,590)GFN Turquoise: Earned 5,000,000 credits (9,675,760)PSA Gold: Earned 500,000 credits (525,639)
Message 127383 - Posted: 24 Feb 2019 | 12:02:28 UTC

Including a primorial like that helps with the numbers not being divisible by the smallest primes.
Check out the full paper that discusses the tricks to find APs fastest here (link from one of the sticky topics in this subforum).

Profile JeppeSNProject donor
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321 LLR Silver: Earned 100,000 credits (360,928)Cullen LLR Bronze: Earned 10,000 credits (98,851)ESP LLR Silver: Earned 100,000 credits (139,922)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Ruby: Earned 2,000,000 credits (2,486,479)PSP LLR Silver: Earned 100,000 credits (212,242)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Bronze: Earned 10,000 credits (16,010)SGS LLR Bronze: Earned 10,000 credits (32,330)TRP LLR Bronze: Earned 10,000 credits (71,060)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve Silver: Earned 100,000 credits (100,631)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 127385 - Posted: 24 Feb 2019 | 12:18:09 UTC

In an arithmetic progression of form An + B, a prime p which does not divide the common difference A, will divide every p-th term of the progression.

For example, in any progression of form 60n + B, no matter what B is, every seventh term is divisible by the prime 7. That is because 7 does not divide 60. So if you want an AP that is greater than 7 in length (or just greater than 6 if no term of the AP equals 7 itself), you want a coefficient A which is divisible by 7.

If you want an AP of length 23, 24, 25, 26, 27, or 28, then the A in the formula An + B must be divisible by all the primes 2, 3, 5, 7, 11, 13, 17, 19, and 23.

This explains the 23# you see.

When, in the future, someone searches for an AP29 (and one not starting from the term 29, so B≠29), they would need to substitute 23# with 29#.

/JeppeSN

DarrylProject donor
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Joined: 21 Mar 09
Posts: 95
ID: 37210
Credit: 336,896,165
RAC: 222,625
Found 1 prime in the 2019 Tour de Primes321 LLR Gold: Earned 500,000 credits (503,464)Cullen LLR Gold: Earned 500,000 credits (500,149)ESP LLR Gold: Earned 500,000 credits (515,497)Generalized Cullen/Woodall LLR Gold: Earned 500,000 credits (527,019)PPS LLR Jade: Earned 10,000,000 credits (12,360,069)PSP LLR Amethyst: Earned 1,000,000 credits (1,006,254)SoB LLR Gold: Earned 500,000 credits (538,748)SR5 LLR Turquoise: Earned 5,000,000 credits (5,387,020)SGS LLR Sapphire: Earned 20,000,000 credits (20,006,208)TRP LLR Gold: Earned 500,000 credits (506,533)Woodall LLR Gold: Earned 500,000 credits (536,003)321 Sieve Jade: Earned 10,000,000 credits (10,011,269)Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (52,021,564)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,134,887)PPS Sieve Double Bronze: Earned 100,000,000 credits (100,260,654)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,761,921)TRP Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,020,224)AP 26/27 Double Bronze: Earned 100,000,000 credits (100,055,458)GFN Sapphire: Earned 20,000,000 credits (20,247,688)
Message 127388 - Posted: 24 Feb 2019 | 13:09:22 UTC

Thanks for both of these answers. It makes much more sense now!

Cheers,
Darryl
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Message boards : AP26 - AP27 Search : 23#

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