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Message boards : General discussion : Fermat Divisor question

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Profile Jordan Romaidis
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Found 1 prime in the 2018 Tour de PrimesFound 2 primes in the 2019 Tour de Primes321 LLR Silver: Earned 100,000 credits (466,293)Cullen LLR Ruby: Earned 2,000,000 credits (2,080,460)ESP LLR Silver: Earned 100,000 credits (362,347)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (109,931)PPS LLR Sapphire: Earned 20,000,000 credits (22,478,646)PSP LLR Bronze: Earned 10,000 credits (23,710)SoB LLR Sapphire: Earned 20,000,000 credits (37,489,376)SR5 LLR Turquoise: Earned 5,000,000 credits (8,600,211)SGS LLR Amethyst: Earned 1,000,000 credits (1,102,412)TRP LLR Silver: Earned 100,000 credits (390,832)Woodall LLR Jade: Earned 10,000,000 credits (15,028,246)321 Sieve Gold: Earned 500,000 credits (843,471)Generalized Cullen/Woodall Sieve Gold: Earned 500,000 credits (511,148)PPS Sieve Sapphire: Earned 20,000,000 credits (22,902,574)AP 26/27 Jade: Earned 10,000,000 credits (15,157,207)GFN Sapphire: Earned 20,000,000 credits (45,227,183)PSA Emerald: Earned 50,000,000 credits (52,891,089)
Message 126970 - Posted: 17 Feb 2019 | 19:16:53 UTC

Can these only be found via PPS LLR?

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 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (2,063,182)Cullen LLR Ruby: Earned 2,000,000 credits (2,005,249)ESP LLR Ruby: Earned 2,000,000 credits (2,001,789)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,115,831)PPS LLR Ruby: Earned 2,000,000 credits (2,768,012)PSP LLR Ruby: Earned 2,000,000 credits (2,632,269)SoB LLR Sapphire: Earned 20,000,000 credits (29,872,580)SR5 LLR Turquoise: Earned 5,000,000 credits (7,691,131)SGS LLR Ruby: Earned 2,000,000 credits (2,011,264)TRP LLR Ruby: Earned 2,000,000 credits (2,433,520)Woodall LLR Ruby: Earned 2,000,000 credits (2,176,414)321 Sieve Gold: Earned 500,000 credits (615,907)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (20,110,788)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 Turquoise: Earned 5,000,000 credits (7,086,053)GFN Emerald: Earned 50,000,000 credits (60,376,816)PSA Jade: Earned 10,000,000 credits (10,038,118)
Message 126971 - Posted: 17 Feb 2019 | 19:34:37 UTC - in response to Message 126970.

Can these only be found via PPS LLR?


Any Proth number can be a Fermat divisor. (k * 2 ^ n + 1)

SGS no.
Woodall no.
GFN no.
321 some yes, some no.
TRP no.
GCW no.
SR5 no.
Cullen yes.
ESP yes.
PSP yes.
SoB yes.
PPSE yes
PPS yes
PPS-MEGA yes.

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Please do not PM me with support questions. Ask on the forums instead. Thank you!

My lucky number is 75898524288+1

Profile Jordan Romaidis
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Joined: 11 May 17
Posts: 120
ID: 880615
Credit: 225,661,500
RAC: 593,696
Found 1 prime in the 2018 Tour de PrimesFound 2 primes in the 2019 Tour de Primes321 LLR Silver: Earned 100,000 credits (466,293)Cullen LLR Ruby: Earned 2,000,000 credits (2,080,460)ESP LLR Silver: Earned 100,000 credits (362,347)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (109,931)PPS LLR Sapphire: Earned 20,000,000 credits (22,478,646)PSP LLR Bronze: Earned 10,000 credits (23,710)SoB LLR Sapphire: Earned 20,000,000 credits (37,489,376)SR5 LLR Turquoise: Earned 5,000,000 credits (8,600,211)SGS LLR Amethyst: Earned 1,000,000 credits (1,102,412)TRP LLR Silver: Earned 100,000 credits (390,832)Woodall LLR Jade: Earned 10,000,000 credits (15,028,246)321 Sieve Gold: Earned 500,000 credits (843,471)Generalized Cullen/Woodall Sieve Gold: Earned 500,000 credits (511,148)PPS Sieve Sapphire: Earned 20,000,000 credits (22,902,574)AP 26/27 Jade: Earned 10,000,000 credits (15,157,207)GFN Sapphire: Earned 20,000,000 credits (45,227,183)PSA Emerald: Earned 50,000,000 credits (52,891,089)
Message 126972 - Posted: 17 Feb 2019 | 19:40:27 UTC - in response to Message 126971.

Thank you Michael!

Yves Gallot
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Message 126974 - Posted: 17 Feb 2019 | 20:06:47 UTC - in response to Message 126971.

GFN no.

25 · 22141884 + 1 divides F2141872 and is a generalized Fermat number.

Profile Michael GoetzProject donor
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Joined: 21 Jan 10
Posts: 12186
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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 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (2,063,182)Cullen LLR Ruby: Earned 2,000,000 credits (2,005,249)ESP LLR Ruby: Earned 2,000,000 credits (2,001,789)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,115,831)PPS LLR Ruby: Earned 2,000,000 credits (2,768,012)PSP LLR Ruby: Earned 2,000,000 credits (2,632,269)SoB LLR Sapphire: Earned 20,000,000 credits (29,872,580)SR5 LLR Turquoise: Earned 5,000,000 credits (7,691,131)SGS LLR Ruby: Earned 2,000,000 credits (2,011,264)TRP LLR Ruby: Earned 2,000,000 credits (2,433,520)Woodall LLR Ruby: Earned 2,000,000 credits (2,176,414)321 Sieve Gold: Earned 500,000 credits (615,907)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (20,110,788)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 Turquoise: Earned 5,000,000 credits (7,086,053)GFN Emerald: Earned 50,000,000 credits (60,376,816)PSA Jade: Earned 10,000,000 credits (10,038,118)
Message 126978 - Posted: 17 Feb 2019 | 21:09:07 UTC - in response to Message 126974.

GFN no.

25 · 22141884 + 1 divides F2141872 and is a generalized Fermat number.



"GFN" meaning our GFN projects.
____________
Please do not PM me with support questions. Ask on the forums instead. Thank you!

My lucky number is 75898524288+1

Profile JeppeSNProject donor
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Joined: 5 Apr 14
Posts: 824
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Credit: 9,981,135
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321 LLR Bronze: Earned 10,000 credits (48,835)Cullen LLR Bronze: Earned 10,000 credits (98,851)ESP LLR Bronze: Earned 10,000 credits (13,226)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Amethyst: Earned 1,000,000 credits (1,791,483)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Bronze: Earned 10,000 credits (16,010)TRP LLR Bronze: Earned 10,000 credits (14,746)Woodall LLR Silver: Earned 100,000 credits (109,455)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 126983 - Posted: 17 Feb 2019 | 21:29:22 UTC - in response to Message 126971.

Can these only be found via PPS LLR?


Any Proth number can be a Fermat divisor. (k * 2 ^ n + 1)


And if the number k*2^n + 1 is a divisor of a Fermat number, say F(m), then the index m satisfies:

m ≤ n-2

For generalized Fermat numbers GF(m, b) and xGF(m, b, a) we can only say

m ≤ n-1

Note that the number k*2^n + 1 is not necessarily a Proth number in the strict sense that 2^n > k. For example the four factors of F(10) = 2^1024 + 1 are:
11131*2^12 + 1 395937*2^14 + 1 1137640572563481089664199400165229051*2^12 + 1 15922836231138695035093355565980212884107486675001451682970617160257863311947248971452664548043591906237644522563833477152239872181860196421948439690685317315553051258143326480945577516888976026564843006895573500498133825643594092555886322403200003*2^13 + 1

and none of them are Proth primes in the strict sense.

/JeppeSN

Yves Gallot
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Message 126988 - Posted: 17 Feb 2019 | 21:50:51 UTC - in response to Message 126978.

GFN no.

25 · 22141884 + 1 divides F2141872 and is a generalized Fermat number.

"GFN" meaning our GFN projects.

A GFN can divide a Fermat number.
10590941048576 + 1 may be the largest known Fermat divisor (with a probability of (2/1059094)1048576).

Profile JeppeSNProject donor
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Joined: 5 Apr 14
Posts: 824
ID: 306875
Credit: 9,981,135
RAC: 3,096
321 LLR Bronze: Earned 10,000 credits (48,835)Cullen LLR Bronze: Earned 10,000 credits (98,851)ESP LLR Bronze: Earned 10,000 credits (13,226)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Amethyst: Earned 1,000,000 credits (1,791,483)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Bronze: Earned 10,000 credits (16,010)TRP LLR Bronze: Earned 10,000 credits (14,746)Woodall LLR Silver: Earned 100,000 credits (109,455)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 126989 - Posted: 17 Feb 2019 | 21:51:05 UTC - in response to Message 126978.

GFN no.

25 · 22141884 + 1 divides F2141872 and is a generalized Fermat number.



"GFN" meaning our GFN projects.


Let me take a recent GFN16 project find as an example, namely the prime:

53942572^65536 + 1

To see this in the Proth form, we would need to split the base 53942572 into a power of two and an odd number. It goes like this 53942572 = 13485643*2^2 and so the GFN16 prime can be written:

13485643^65536*(2^2)^65536 + 1

or:

(13485643^65536)*2^131072 + 1

So this is of the "Proth" form k*2^n + 1 where n=131072 and k is the enormous odd number 13485643^65536. So given the form, it could divide some F(m) with m ≤ 131070. However, since k is so incredibly huge, there is no need of checking it, the chance is too slim.

If we found a GFN16 prime b^65536+1 with the special property that b is a really small odd number times a big power of two, the chances would be a little bit better. Say, if 25165824^65536+1 were prime (I have not checked, so it probably is not), then 25165824 = 3*2^23, and we would end up with "Proth" form:

3^65536*(2^23)^65536 + 1

or:

(3^65536)*2^1507328 + 1

but even in this extreme case, the size of k (= 3^65536) is so big that it is not worth the computation time to check if it is a Fermat divisor.

/JeppeSN

Profile JeppeSNProject donor
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Joined: 5 Apr 14
Posts: 824
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Credit: 9,981,135
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321 LLR Bronze: Earned 10,000 credits (48,835)Cullen LLR Bronze: Earned 10,000 credits (98,851)ESP LLR Bronze: Earned 10,000 credits (13,226)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Amethyst: Earned 1,000,000 credits (1,791,483)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Bronze: Earned 10,000 credits (16,010)TRP LLR Bronze: Earned 10,000 credits (14,746)Woodall LLR Silver: Earned 100,000 credits (109,455)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 126992 - Posted: 17 Feb 2019 | 22:01:33 UTC

Similar remarks apply to GCW primes of the generalized Cullen form if the base is even, such as:

1323365*116^1323365 + 1 (Scott Brown, January 2018)

which could in principle be a Fermat divisor, but it is a waste of effort to check for it.

/JeppeSN

Profile JeppeSNProject donor
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Joined: 5 Apr 14
Posts: 824
ID: 306875
Credit: 9,981,135
RAC: 3,096
321 LLR Bronze: Earned 10,000 credits (48,835)Cullen LLR Bronze: Earned 10,000 credits (98,851)ESP LLR Bronze: Earned 10,000 credits (13,226)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Amethyst: Earned 1,000,000 credits (1,791,483)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Bronze: Earned 10,000 credits (16,010)TRP LLR Bronze: Earned 10,000 credits (14,746)Woodall LLR Silver: Earned 100,000 credits (109,455)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 126996 - Posted: 17 Feb 2019 | 22:21:43 UTC

Shorter put, I agree with Yves.

Note that his example 25*2^2141884 + 1 was found by "our" PPS LLR search, and was originally on the GFN Top 20.

/JeppeSN

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Message boards : General discussion : Fermat Divisor question

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