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Message boards : Wieferich and Wall-Sun-Sun Prime Search : We found a near-Wieferich with |A| ≤ 10

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Profile JeppeSNProject donor
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Joined: 5 Apr 14
Posts: 1785
ID: 306875
Credit: 48,177,424
RAC: 13,351
Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (593,283)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (174,818)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (112,799)PPS LLR Jade: Earned 10,000,000 credits (18,505,516)PSP LLR Silver: Earned 100,000 credits (428,457)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (210,142)SGS LLR Silver: Earned 100,000 credits (112,277)TRP LLR Silver: Earned 100,000 credits (476,246)Woodall LLR Silver: Earned 100,000 credits (281,400)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)GFN Ruby: Earned 2,000,000 credits (4,636,807)WW Jade: Earned 10,000,000 credits (13,756,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 149760 - Posted: 30 Mar 2021 | 22:37:14 UTC

Dear all,

User tng and double checker Grzegorz Roman Granowski just found the following:

3156824277937156367 is a Wieferich special instance (+1 +7 p)
(about 3*10^18).

This A=+7 is the lowest |A| seen since p=2276306935816523 (about 2*10^15, in the year 2006). There is a list of near-Wieferichs with |A| at most 10 on Wikipedia, Near-Wieferich primes, and there is a list of near-Wieferichs with |A| strictly below 10 [in the LINKS section] on OEIS A246568.

Congratulations to tng!

An overview of new WW finds is on WW Statistics.

/JeppeSN

Profile Eudy SilvaProject donor
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Discovered 3 mega primesFound 2 primes in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de PrimesFound 1 prime in the 2021 Tour de Primes321 LLR Amethyst: Earned 1,000,000 credits (1,187,213)Cullen LLR Amethyst: Earned 1,000,000 credits (1,349,947)ESP LLR Amethyst: Earned 1,000,000 credits (1,151,976)Generalized Cullen/Woodall LLR Amethyst: Earned 1,000,000 credits (1,199,207)PPS LLR Sapphire: Earned 20,000,000 credits (30,057,152)PSP LLR Amethyst: Earned 1,000,000 credits (1,476,639)SoB LLR Ruby: Earned 2,000,000 credits (2,432,091)SR5 LLR Ruby: Earned 2,000,000 credits (4,158,377)SGS LLR Amethyst: Earned 1,000,000 credits (1,462,951)TRP LLR Ruby: Earned 2,000,000 credits (2,010,675)Woodall LLR Amethyst: Earned 1,000,000 credits (1,231,973)321 Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,119,749)Generalized Cullen/Woodall Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,005,331)PPS Sieve Double Bronze: Earned 100,000,000 credits (100,101,845)AP 26/27 Double Bronze: Earned 100,000,000 credits (100,120,852)GFN Double Bronze: Earned 100,000,000 credits (122,249,253)WW Double Silver: Earned 200,000,000 credits (200,192,000)PSA Ruby: Earned 2,000,000 credits (2,059,222)
Message 149761 - Posted: 30 Mar 2021 | 23:13:49 UTC - in response to Message 149760.

Congrats to tng and Grzegorz Roman Granowski !
____________

"Accidit in puncto, quod non contingit in anno."
Something that does not occur in a year may, perchance, happen in a moment.

Ravi Fernando
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321 LLR Amethyst: Earned 1,000,000 credits (1,061,204)Cullen LLR Silver: Earned 100,000 credits (111,878)ESP LLR Bronze: Earned 10,000 credits (16,570)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (68,801)PPS LLR Ruby: Earned 2,000,000 credits (4,609,657)PSP LLR Silver: Earned 100,000 credits (106,263)SoB LLR Silver: Earned 100,000 credits (258,849)SR5 LLR Bronze: Earned 10,000 credits (76,537)SGS LLR Silver: Earned 100,000 credits (148,878)TRP LLR Silver: Earned 100,000 credits (254,161)Woodall LLR Bronze: Earned 10,000 credits (97,444)321 Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,001,667)AP 26/27 Bronze: Earned 10,000 credits (72,774)GFN Amethyst: Earned 1,000,000 credits (1,170,348)WW Bronze: Earned 10,000 credits (12,000)
Message 149763 - Posted: 31 Mar 2021 | 3:04:23 UTC - in response to Message 149760.

This A=+7 is the lowest |A| seen since p=2276306935816523 (about 2*10^15, in the year 2006).

In particular, the ratio |A|/p ~ 2.22 * 10^-18 is easily the lowest yet for a near-Wieferich, excluding the two known Wieferich primes of course. The previous record for this was 40/1713380095653669769 ~ 1.27 * 10^-17, found by none other than Grzegorz Roman Granowski early last month.

Profile JeppeSNProject donor
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Joined: 5 Apr 14
Posts: 1785
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Credit: 48,177,424
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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (593,283)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (174,818)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (112,799)PPS LLR Jade: Earned 10,000,000 credits (18,505,516)PSP LLR Silver: Earned 100,000 credits (428,457)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (210,142)SGS LLR Silver: Earned 100,000 credits (112,277)TRP LLR Silver: Earned 100,000 credits (476,246)Woodall LLR Silver: Earned 100,000 credits (281,400)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)GFN Ruby: Earned 2,000,000 credits (4,636,807)WW Jade: Earned 10,000,000 credits (13,756,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 149765 - Posted: 31 Mar 2021 | 7:26:42 UTC - in response to Message 149763.
Last modified: 31 Mar 2021 | 7:35:02 UTC

This A=+7 is the lowest |A| seen since p=2276306935816523 (about 2*10^15, in the year 2006).

In particular, the ratio |A|/p ~ 2.22 * 10^-18 is easily the lowest yet for a near-Wieferich, excluding the two known Wieferich primes of course. The previous record for this was 40/1713380095653669769 ~ 1.27 * 10^-17, found by none other than Grzegorz Roman Granowski early last month.

Yes, maybe you could make a Wieferich version of my near-Wall-Sun-Sun sequence OEIS A339855. In one possible version of it, it would end at the first Wieferich prime:
3, 5, 7, 17, 31, 59, 71, 251, 379, 1093.


But you could also make a version that started from 3 but ignored true Wieferich primes:
3, 5, 7, 17, 31, 59, 71, 251, 379, 1733, 2633, 2659, 15823, 60631, 352691, ...

Or you could make a version that started from just after the second Wieferich prime 3511 and ended at the third Wieferich prime (if it exists):
3517, 3527, 3541, 3923, 4723, 5179, 9419, 13463, 15329, 15823, 60631, 352691, ...


In fact, you could probably submit all these three if you wanted. Do any of you have any opinion on that?

There is another question with these near-Wieferichs, because there are two conventions on how to measure nearness. PrimeGrid (like some of its predecessors) looks at 2^{(p-1)/2} modulo p^2, and defines A based on that. But it is maybe more natural to include the final squaring and instead consider 2^{p-1} modulo p^2, and if you read the Wikipedia section I linked, this was done by Dorais & Klyve, and gives another list of near-Wieferichs.

/JeppeSN

Profile Rick Reynolds
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Discovered 3 mega primesFound 2 primes in the 2018 Tour de PrimesFound 2 primes in the 2019 Tour de PrimesFound 12 primes in the 2020 Tour de PrimesFound 2 primes in the 2021 Tour de PrimesFound 1 mega prime in the 2021 Tour de Primes321 LLR Jade: Earned 10,000,000 credits (10,244,909)Cullen LLR Jade: Earned 10,000,000 credits (10,194,451)ESP LLR Jade: Earned 10,000,000 credits (10,735,017)Generalized Cullen/Woodall LLR Jade: Earned 10,000,000 credits (10,216,354)PPS LLR Emerald: Earned 50,000,000 credits (73,178,392)PSP LLR Turquoise: Earned 5,000,000 credits (5,151,448)SoB LLR Jade: Earned 10,000,000 credits (10,045,557)SR5 LLR Jade: Earned 10,000,000 credits (14,871,275)SGS LLR Turquoise: Earned 5,000,000 credits (7,860,522)TRP LLR Jade: Earned 10,000,000 credits (11,130,742)Woodall LLR Turquoise: Earned 5,000,000 credits (5,419,909)321 Sieve (suspended) Double Bronze: Earned 100,000,000 credits (101,538,236)Cullen/Woodall Sieve (suspended) Double Bronze: Earned 100,000,000 credits (127,922,665)Generalized Cullen/Woodall Sieve (suspended) Sapphire: Earned 20,000,000 credits (30,984,158)PPS Sieve Double Ruby: Earned 2,000,000,000 credits (2,840,173,136)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Turquoise: Earned 5,000,000 credits (8,134,964)TRP Sieve (suspended) Turquoise: Earned 5,000,000 credits (7,013,696)AP 26/27 Double Silver: Earned 200,000,000 credits (276,723,135)GFN Double Silver: Earned 200,000,000 credits (349,277,565)WW Double Amethyst: Earned 1,000,000,000 credits (1,431,508,000)PSA Double Amethyst: Earned 1,000,000,000 credits (1,370,700,486)
Message 149770 - Posted: 31 Mar 2021 | 12:20:30 UTC
Last modified: 31 Mar 2021 | 12:31:21 UTC

Congrats tng!

Grzegorz Roman GranowskiProject donor
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Discovered 1 Fermat divisor321 LLR Amethyst: Earned 1,000,000 credits (1,405,414)Cullen LLR Silver: Earned 100,000 credits (465,280)ESP LLR Turquoise: Earned 5,000,000 credits (5,422,341)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (38,023)PPS LLR Double Silver: Earned 200,000,000 credits (245,402,626)PSP LLR Emerald: Earned 50,000,000 credits (64,720,207)SoB LLR Double Gold: Earned 500,000,000 credits (549,463,742)SR5 LLR Emerald: Earned 50,000,000 credits (72,241,221)TRP LLR Gold: Earned 500,000 credits (935,975)Woodall LLR Gold: Earned 500,000 credits (740,422)321 Sieve (suspended) Silver: Earned 100,000 credits (461,396)Cullen/Woodall Sieve (suspended) Silver: Earned 100,000 credits (189,614)PPS Sieve Double Ruby: Earned 2,000,000,000 credits (3,464,910,945)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Silver: Earned 100,000 credits (213,104)TRP Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,091,635)AP 26/27 Double Silver: Earned 200,000,000 credits (479,807,068)GFN Double Turquoise: Earned 5,000,000,000 credits (6,039,335,039)WW Double Jade: Earned 10,000,000,000 credits (14,779,300,000)
Message 149771 - Posted: 31 Mar 2021 | 13:40:42 UTC - in response to Message 149761.

Congrats to tng and Grzegorz Roman Granowski !



Thank You ...

my ... power of gpu is going to be updated soon ... and hopefully ...

we'll find a = 0 ... SOON ...

best regards, Grzegorz Roman Granowski
____________

Kellen
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Credit: 1,600,003,090
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Discovered 2 mega primes321 LLR Sapphire: Earned 20,000,000 credits (20,008,344)Cullen LLR Sapphire: Earned 20,000,000 credits (20,000,917)ESP LLR Sapphire: Earned 20,000,000 credits (20,001,144)Generalized Cullen/Woodall LLR Sapphire: Earned 20,000,000 credits (20,011,942)PPS LLR Sapphire: Earned 20,000,000 credits (20,000,023)PSP LLR Sapphire: Earned 20,000,000 credits (20,008,714)SoB LLR Sapphire: Earned 20,000,000 credits (20,035,621)SR5 LLR Sapphire: Earned 20,000,000 credits (20,004,459)SGS LLR Sapphire: Earned 20,000,000 credits (20,000,002)TRP LLR Sapphire: Earned 20,000,000 credits (20,000,236)Woodall LLR Sapphire: Earned 20,000,000 credits (20,017,137)321 Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,000,569)Generalized Cullen/Woodall Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,006,424)PPS Sieve Sapphire: Earned 20,000,000 credits (49,900,913)AP 26/27 Sapphire: Earned 20,000,000 credits (45,002,633)GFN Double Silver: Earned 200,000,000 credits (200,000,012)WW Sapphire: Earned 20,000,000 credits (45,004,000)PSA Double Amethyst: Earned 1,000,000,000 credits (1,000,000,000)
Message 149772 - Posted: 31 Mar 2021 | 14:18:27 UTC - in response to Message 149765.

Or you could make a version that started from just after the second Wieferich prime 3511 and ended at the third Wieferich prime (if it exists):
3517, 3527, 3541, 3923, 4723, 5179, 9419, 13463, 15329, 15823, 60631, 352691, ...


In fact, you could probably submit all these three if you wanted. Do any of you have any opinion on that?


I like this one the most. Ignoring the two initial Wieferich primes seems like a reasonable thing to do, given how close they are to zero. From here, I think that the interesting property is the gap between subsequent terms.

A fourth option though would be to include all terms and "reset" at each Wieferich prime.
3, 5, 7, 17, 31, 59, 71, 251, 379, 1093, *terms between 1093 and 3511*, 3511, 3517, 3527, 3541, 3923, 4723, 5179...

Not sure if OEIS accepts resetting sequences at specific terms though.

Congrats to tng and Grzegorz Roman Granowski for the excellent find!

Message boards : Wieferich and Wall-Sun-Sun Prime Search : We found a near-Wieferich with |A| ≤ 10

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