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Message boards : General discussion : Proposed new formula

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Profile crowattorney
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Message 140511 - Posted: 31 May 2020 | 0:01:09 UTC

Hello! I am relatively new to primegrid and I want to run a formula not listed on here,
3^k + 2
So far I have not been able to see any documentation on it anywhere.
Should I ask to get it running or should I go somewhere else?
____________
My lucky number is 3504206+2

Profile dannyridel
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321 LLR Silver: Earned 100,000 credits (100,228)Cullen LLR Bronze: Earned 10,000 credits (36,962)ESP LLR Bronze: Earned 10,000 credits (36,309)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (22,645)PPS LLR Silver: Earned 100,000 credits (219,290)PSP LLR Silver: Earned 100,000 credits (100,559)SoB LLR Silver: Earned 100,000 credits (138,888)SR5 LLR Bronze: Earned 10,000 credits (64,280)SGS LLR Bronze: Earned 10,000 credits (35,164)TRP LLR Bronze: Earned 10,000 credits (67,784)Woodall LLR Bronze: Earned 10,000 credits (40,976)321 Sieve Gold: Earned 500,000 credits (506,814)Generalized Cullen/Woodall Sieve (suspended) Bronze: Earned 10,000 credits (30,033)PPS Sieve Gold: Earned 500,000 credits (825,895)AP 26/27 Silver: Earned 100,000 credits (141,505)GFN Gold: Earned 500,000 credits (859,368)PSA Silver: Earned 100,000 credits (373,034)
Message 140512 - Posted: 31 May 2020 | 2:29:59 UTC

I'm not sure if you're prepared well enough, but you could get advice and start your own search here
____________
SHSID Electronics Group
SHSIDElectronicsGroup@outlook.com

GFN-14: 50103906^16384+1
Proth "SoB": 44243*2^440969+1


Kellen
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Discovered 2 mega primes321 LLR Amethyst: Earned 1,000,000 credits (1,046,831)Cullen LLR Ruby: Earned 2,000,000 credits (4,120,978)ESP LLR Ruby: Earned 2,000,000 credits (3,231,650)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (3,649,553)PPS LLR Jade: Earned 10,000,000 credits (17,577,050)PSP LLR Amethyst: Earned 1,000,000 credits (1,620,415)SoB LLR Jade: Earned 10,000,000 credits (11,021,129)SR5 LLR Turquoise: Earned 5,000,000 credits (5,127,306)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 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 (42,292,566)AP 26/27 Sapphire: Earned 20,000,000 credits (40,001,442)GFN Emerald: Earned 50,000,000 credits (95,192,575)PSA Double Amethyst: Earned 1,000,000,000 credits (1,000,000,000)
Message 140514 - Posted: 31 May 2020 | 2:50:28 UTC - in response to Message 140511.

Hello! I am relatively new to primegrid and I want to run a formula not listed on here,
3^k + 2
So far I have not been able to see any documentation on it anywhere.
Should I ask to get it running or should I go somewhere else?


Hi R. H. Alderton,

You have asked my favorite kind of question! The kind of question that starts with "I wonder..." I ask these quite often :)

Here are the first few values of k which result in 3k+2 being prime:

1 2 3 4 8 10 14 15 24 26 36 63 98 110 123 126 139 235 243 315 363 386 391 494 1131 1220 1503 1858


These are pretty tiny primes, with the last one on the list having 887 digits, but unfortunately things really slow down past that point when searching for more (at least with Pari-GP) because there are no quick and special ways to test primality for a number of that form (to the best of my knowledge), like there are with other forms like k3n+1 or any of the other prime forms searched here. You can read about the tests that can be performed when a candidate prime is of the "+1" variety here: https://primes.utm.edu/prove/prove3_1.html

It would be possible to keep searching for more, but it would be a painstaking search I think. Good luck if you do try though and keep us posted with any more finds!

Regards,
Kellen

Ravi Fernando
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Message 140515 - Posted: 31 May 2020 | 6:47:03 UTC - in response to Message 140514.

Here is a list of all known primes (or probable primes) of this form. As Kellen said, we currently have no practical way to prove that a really large number of this form is actually prime, hence why these are only "probable primes" beyond k=10988.

Profile crowattorney
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Cullen LLR Silver: Earned 100,000 credits (146,809)PPS LLR Silver: Earned 100,000 credits (306,055)SGS LLR Silver: Earned 100,000 credits (159,774)321 Sieve Ruby: Earned 2,000,000 credits (2,227,910)PPS Sieve Bronze: Earned 10,000 credits (80,904)PSA Bronze: Earned 10,000 credits (14,644)
Message 140520 - Posted: 31 May 2020 | 16:13:40 UTC

Hello, and thank you for the advice,
couldn't we just apply Fermat's little theorem to it?
I know the numbers would reach immense scales very quickly
2^[3^504207 + 2] - 2 =~10^10^10^5.04
But hypothetically it could be done, right?
____________
My lucky number is 3504206+2

Ravi Fernando
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Joined: 21 Mar 19
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321 LLR Silver: Earned 100,000 credits (434,563)Cullen LLR Bronze: Earned 10,000 credits (82,217)ESP LLR Bronze: Earned 10,000 credits (16,570)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (12,551)PPS LLR Ruby: Earned 2,000,000 credits (2,123,888)PSP LLR Bronze: Earned 10,000 credits (26,371)SoB LLR Silver: Earned 100,000 credits (183,524)SR5 LLR Bronze: Earned 10,000 credits (59,307)SGS LLR Silver: Earned 100,000 credits (148,878)TRP LLR Silver: Earned 100,000 credits (195,905)Woodall LLR Bronze: Earned 10,000 credits (19,925)321 Sieve Turquoise: Earned 5,000,000 credits (5,001,667)AP 26/27 Bronze: Earned 10,000 credits (72,774)GFN Silver: Earned 100,000 credits (110,081)
Message 140522 - Posted: 31 May 2020 | 18:13:41 UTC - in response to Message 140520.

We can certainly use Fermat's little theorem--but Fermat's little theorem never tells us that something is guaranteed to be prime, only that if p is prime, then a^(p-1) is congruent to 1 mod p for all integers a not divisible by p. The converse is usually true, but not always, because of Carmichael numbers (e.g. 561) and other pseudoprimes (e.g. 341 in base a=2).

So if we test Fermat's congruence for a candidate p with several different choices of a, we can be highly confident, but not 100% confident, that it's prime. In fact, when we call something a "probable prime", that usually means that we've done these tests, perhaps along with some trial factoring and a few souped-up versions of the same idea (e.g. Miller-Rabin or BPSW) for extra confidence.

One more note: since (as you said) these tests involve really large numbers, we have to be slightly careful when implementing them. The number 2^(3^504207 + 2) - 1 has far more digits than there are atoms in the universe, so there is no hope of representing it in binary on any physical computer. But fortunately we don't need to: we only need its congruence class modulo the 240568-digit number p = 3^504207 + 2. We can do all our calculations mod p, and some modular exponentiation tricks make it possible to compute 2^(3^504207 + 2) - 1 mod p with only about a million multiplication steps. This is feasible on modern computers.

Profile crowattorney
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Cullen LLR Silver: Earned 100,000 credits (146,809)PPS LLR Silver: Earned 100,000 credits (306,055)SGS LLR Silver: Earned 100,000 credits (159,774)321 Sieve Ruby: Earned 2,000,000 credits (2,227,910)PPS Sieve Bronze: Earned 10,000 credits (80,904)PSA Bronze: Earned 10,000 credits (14,644)
Message 140525 - Posted: 1 Jun 2020 | 2:50:55 UTC - in response to Message 140522.

And even if it turned out to not be prime but satisfied the tests, we have a new Carmichael number!
____________
My lucky number is 3504206+2

Profile JeppeSNProject donor
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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (529,293)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (139,922)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Turquoise: Earned 5,000,000 credits (9,594,179)PSP LLR Silver: Earned 100,000 credits (212,242)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Silver: Earned 100,000 credits (145,419)SGS LLR Silver: Earned 100,000 credits (105,212)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve 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 Amethyst: Earned 1,000,000 credits (1,674,106)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 140540 - Posted: 1 Jun 2020 | 22:11:16 UTC

So it turns out the formula is not new. Ravi mentioned OEIS A051783 already. If you could find any new members, you could contribute to that sequence.

But there is also PRP Top. See PRP Top search for 3^n+2. If you found any new ones, you would submit them there.

Since you are not going to be able to prove these numbers prime, they are not going to be admitted into the Prime Pages lists.

/JeppeSN

Kellen
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Discovered 2 mega primes321 LLR Amethyst: Earned 1,000,000 credits (1,046,831)Cullen LLR Ruby: Earned 2,000,000 credits (4,120,978)ESP LLR Ruby: Earned 2,000,000 credits (3,231,650)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (3,649,553)PPS LLR Jade: Earned 10,000,000 credits (17,577,050)PSP LLR Amethyst: Earned 1,000,000 credits (1,620,415)SoB LLR Jade: Earned 10,000,000 credits (11,021,129)SR5 LLR Turquoise: Earned 5,000,000 credits (5,127,306)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 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 (42,292,566)AP 26/27 Sapphire: Earned 20,000,000 credits (40,001,442)GFN Emerald: Earned 50,000,000 credits (95,192,575)PSA Double Amethyst: Earned 1,000,000,000 credits (1,000,000,000)
Message 140986 - Posted: 22 Jun 2020 | 22:10:34 UTC

Just noticed this pop up on PrimePages: https://primes.utm.edu/primes/page.php?id=130986

Looks like someone is working to prove primality for these :)

Profile JeppeSNProject donor
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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (529,293)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (139,922)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Turquoise: Earned 5,000,000 credits (9,594,179)PSP LLR Silver: Earned 100,000 credits (212,242)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Silver: Earned 100,000 credits (145,419)SGS LLR Silver: Earned 100,000 credits (105,212)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve 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 Amethyst: Earned 1,000,000 credits (1,674,106)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 141020 - Posted: 23 Jun 2020 | 15:59:06 UTC - in response to Message 140986.

Just noticed this pop up on PrimePages: https://primes.utm.edu/primes/page.php?id=130986

Looks like someone is working to prove primality for these :)

This is our Gelly; was discussed on Discord. /JeppeSN

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Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 2 primes in the 2020 Tour de Primes321 LLR Bronze: Earned 10,000 credits (38,954)ESP LLR Gold: Earned 500,000 credits (942,185)PPS LLR Double Bronze: Earned 100,000,000 credits (132,754,154)PSP LLR Silver: Earned 100,000 credits (489,641)SoB LLR Jade: Earned 10,000,000 credits (12,960,428)SR5 LLR Jade: Earned 10,000,000 credits (10,694,695)SGS LLR Gold: Earned 500,000 credits (563,819)TRP LLR Jade: Earned 10,000,000 credits (12,592,743)321 Sieve Silver: Earned 100,000 credits (395,205)PPS Sieve Double Silver: Earned 200,000,000 credits (229,811,183)TRP Sieve (suspended) Gold: Earned 500,000 credits (669,191)AP 26/27 Jade: Earned 10,000,000 credits (15,039,960)GFN Emerald: Earned 50,000,000 credits (52,138,286)
Message 141054 - Posted: 23 Jun 2020 | 22:43:13 UTC - in response to Message 140986.

Just noticed this pop up on PrimePages: https://primes.utm.edu/primes/page.php?id=130986

Looks like someone is working to prove primality for these :)


Oh hey, that's mine! Ravi Fernando recommended it as a nice way to break into the top 20 with a good ECPP machine, so I went with it.

As was discussed, the best chance does really seem to be ECPP for primes of this form. I was a little excited at the prospect of how factorizeable 3^k + 1 and 3^k + 3 were, and using an (N+1)(N-1) method for proving them in general, but considering you get many factors in the thousand digit range that will likely never be factorized in our lifetime, it sort of died there.

Perhaps if you were super-ultra-lucky with a k that both is very factorizeable, leads to a PRP for 3^k +2, and you got extremely lucky with the factors of 3^k+1 and 3^k+3, you could eke out one super-lucky proveable prime that doesn't need a general deterministic primality proof.

It'd be radical if that'd ever happen, but for now, ECPP does a treat on anything smaller than 40k digits with the right hardware.

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Message boards : General discussion : Proposed new formula

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