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Message boards : News : 321 Mega Prime! (2015 Edition, part 2)

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Scott BrownProject donor
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Discovered the World's First base 116 Generalized Cullen prime!!!Discovered 55 mega primesEliminated 7 conjecture "k"sDiscovered 1 Sophie Germain pairDiscovered 2 Fermat divisors2012 Tour de Primes highest prime count2012 Tour de Primes most Mountain Stage primes2015 Tour de Primes highest prime count2016 Tour de Primes highest prime countFound 23 primes in the 2018 Tour de PrimesFound 1 mega prime in the 2018 Tour de PrimesFound 2 primes in the 2018 Tour de Primes Mountain Stage2019 Tour de Primes highest prime countFound 22 primes in the 2019 Tour de Primes2020 Tour de Primes highest prime scoreFound 21 primes in the 2020 Tour de PrimesFound 4 mega primes in the 2020 Tour de PrimesFound 8 primes in the 2021 Tour de PrimesFound 1 mega prime in the 2021 Tour de PrimesFound 21 primes in the 2022 Tour de PrimesFound 1 mega prime in the 2022 Tour de PrimesFound 2 primes in the 2022 Tour de Primes Mountain Stage321 LLR Double Silver: Earned 200,000,000 credits (222,511,049)Cullen LLR Double Silver: Earned 200,000,000 credits (201,409,561)ESP LLR Double Silver: Earned 200,000,000 credits (214,825,147)Generalized Cullen/Woodall LLR Double Silver: Earned 200,000,000 credits (204,939,647)PPS LLR Double Amethyst: Earned 1,000,000,000 credits (1,000,849,509)PSP LLR Double Silver: Earned 200,000,000 credits (201,749,918)SoB LLR Double Silver: Earned 200,000,000 credits (207,028,068)SR5 LLR Double Silver: Earned 200,000,000 credits (245,826,868)SGS LLR Double Silver: Earned 200,000,000 credits (217,421,273)TPS LLR (retired) Silver: Earned 100,000 credits (235,439)TRP LLR Double Silver: Earned 200,000,000 credits (225,190,162)Woodall LLR Double Silver: Earned 200,000,000 credits (201,651,344)321 Sieve (suspended) Double Silver: Earned 200,000,000 credits (235,451,253)Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (83,794,448)Generalized Cullen/Woodall Sieve (suspended) Double Silver: Earned 200,000,000 credits (285,139,652)PPS Sieve Double Ruby: Earned 2,000,000,000 credits (2,895,962,931)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Double Silver: Earned 200,000,000 credits (203,523,358)TRP Sieve (suspended) Double Silver: Earned 200,000,000 credits (201,489,157)AP 26/27 Double Gold: Earned 500,000,000 credits (512,622,591)GFN Double Ruby: Earned 2,000,000,000 credits (2,497,975,933)WW Double Ruby: Earned 2,000,000,000 credits (4,566,068,000)PSA Double Silver: Earned 200,000,000 credits (259,058,048)
Message 86665 - Posted: 12 Jul 2015 | 19:31:33 UTC

On 23 June 2015 20:49:21 UTC, PrimeGrid’s 321 Prime Search project found the mega prime:

3*2^11895718-1

The prime is 3,580,969 digits long and enters Chris Caldwell's The Largest Known Primes Database ranked 12th overall. This is the largest known 321 mega prime and the largest prime found to date at PrimeGrid!

The discovery was made by Michael Schulz (Michael Schulz) of Germany using an Intel(R) Core(TM) i5-4570 CPU @ 3.20GHz with 16 GB RAM running Darwin 14.3.0. This computer took about 9 hours and 57 minutes to complete the primality test using LLR.

The prime was verified on 27 June 2015 3:56:03 UTC, by user Matthew of New Zealand using an Intel(R) Core(TM) i7-4510U CPU @ 2.00GHz with 8 GB RAM running Windows 7 Professional. This computer took about 33 hours and 40 minutes to complete the primality test using LLR.

For more details, please see the official announcement.

Profile JeppeSNProject donor
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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 (15,983,601)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 (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)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 (4,228,147)WW Turquoise: Earned 5,000,000 credits (9,640,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 86667 - Posted: 12 Jul 2015 | 22:15:35 UTC - in response to Message 86665.


For more details, please see the official announcement.


This is a "minus one" type prime.

We can write:

3*2^11895718 - 1 = 3*(2^11895717-1) * 2 + 1


to force it onto "plus" form.

In particular, this prime is -1 (mod 4), not +1 (mod 4).

With knowledge on what factors of (generalized) Fermat numbers look like, it is clear that this prime could not be one. So the section on Fermat divisibility in the above announcement must be a copy-paste mistake.

/JeppeSN

Christopher Siegert
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321 LLR Amethyst: Earned 1,000,000 credits (1,041,657)Cullen LLR Amethyst: Earned 1,000,000 credits (1,366,558)PPS LLR Sapphire: Earned 20,000,000 credits (22,177,547)PSP LLR Amethyst: Earned 1,000,000 credits (1,521,444)SoB LLR Amethyst: Earned 1,000,000 credits (1,751,070)SR5 LLR Amethyst: Earned 1,000,000 credits (1,100,031)SGS LLR Amethyst: Earned 1,000,000 credits (1,014,953)TRP LLR Ruby: Earned 2,000,000 credits (3,647,371)Woodall LLR Amethyst: Earned 1,000,000 credits (1,479,507)Cullen/Woodall Sieve (suspended) Bronze: Earned 10,000 credits (41,181)PPS Sieve Sapphire: Earned 20,000,000 credits (29,702,366)TRP Sieve (suspended) Silver: Earned 100,000 credits (206,682)GFN Jade: Earned 10,000,000 credits (11,960,980)
Message 86670 - Posted: 13 Jul 2015 | 4:09:49 UTC
Last modified: 13 Jul 2015 | 4:16:52 UTC

3*2^11895718 - 1 = 3*(2^11895717-1) * 2 + 1


I'm confused.

Doesn't 3*(2^11895717-1) * 2 + 1 = 3*2^11895718 - 5

It seems you're off by 4.

I think you mean: (3*2^11895717 - 1) * 2 + 1

However, this is not a Proth number as k > 2^n

Profile Michael GoetzProject donor
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The "Shut up already!" badge:  This loud mouth has mansplained on the forums over 10 thousand times!  Sheesh!!!Discovered the World's First GFN-19 prime!!!Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de PrimesFound 2 primes in the 2021 Tour de PrimesFound 2 primes in the 2022 Tour de PrimesFound 1 mega prime in the 2022 Tour de PrimesFound 1 prime in the 2022 Tour de Primes Mountain Stage321 LLR Turquoise: Earned 5,000,000 credits (6,638,389)Cullen LLR Turquoise: Earned 5,000,000 credits (5,038,114)ESP LLR Turquoise: Earned 5,000,000 credits (6,177,890)Generalized Cullen/Woodall LLR Turquoise: Earned 5,000,000 credits (5,094,541)PPS LLR Sapphire: Earned 20,000,000 credits (23,416,635)PSP LLR Turquoise: Earned 5,000,000 credits (7,956,186)SoB LLR Sapphire: Earned 20,000,000 credits (36,067,618)SR5 LLR Jade: Earned 10,000,000 credits (12,645,567)SGS LLR Turquoise: Earned 5,000,000 credits (5,036,951)TRP LLR Turquoise: Earned 5,000,000 credits (5,084,329)Woodall LLR Turquoise: Earned 5,000,000 credits (5,032,821)321 Sieve (suspended) Jade: Earned 10,000,000 credits (10,061,196)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (22,885,121)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,035,522)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,051,121)AP 26/27 Jade: Earned 10,000,000 credits (10,902,645)GFN Emerald: Earned 50,000,000 credits (92,455,703)WW Emerald: Earned 50,000,000 credits (65,888,000)PSA Jade: Earned 10,000,000 credits (12,445,029)
Message 86672 - Posted: 13 Jul 2015 | 6:05:35 UTC

Yes, the PFGW and Fermat divisor part of the announcement was a copy and paste error.
____________
My lucky number is 75898524288+1

Profile JeppeSNProject donor
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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 (15,983,601)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 (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)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 (4,228,147)WW Turquoise: Earned 5,000,000 credits (9,640,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 86681 - Posted: 13 Jul 2015 | 21:21:47 UTC - in response to Message 86670.

3*2^11895718 - 1 = 3*(2^11895717-1) * 2 + 1


I'm confused.

Doesn't 3*(2^11895717-1) * 2 + 1 = 3*2^11895718 - 5

It seems you're off by 4.

I think you mean: (3*2^11895717 - 1) * 2 + 1

However, this is not a Proth number as k > 2^n


Thank you very much, you are entirely correct. That was a typo.

The important thing was that the prime here, when written as k*2^n + 1, must have exponent n at most 1 (exactly n=1 if we require k odd), so it cannot divide a (generalized) Fermat GF(m, b) where m>0.

/JeppeSN

Message boards : News : 321 Mega Prime! (2015 Edition, part 2)

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