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Message boards : General discussion : b^n - b^m - 1 primes

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Posts: 399
ID: 1241833
Credit: 157,022,093
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321 LLR Amethyst: Earned 1,000,000 credits (1,058,073)Cullen LLR Amethyst: Earned 1,000,000 credits (1,169,946)ESP LLR Amethyst: Earned 1,000,000 credits (1,093,381)Generalized Cullen/Woodall LLR Amethyst: Earned 1,000,000 credits (1,148,593)PPS LLR Amethyst: Earned 1,000,000 credits (1,225,852)PSP LLR Amethyst: Earned 1,000,000 credits (1,248,861)SoB LLR Amethyst: Earned 1,000,000 credits (1,669,219)SR5 LLR Amethyst: Earned 1,000,000 credits (1,060,324)SGS LLR Amethyst: Earned 1,000,000 credits (1,108,160)TRP LLR Amethyst: Earned 1,000,000 credits (1,039,866)Woodall LLR Amethyst: Earned 1,000,000 credits (1,129,385)321 Sieve (suspended) Ruby: Earned 2,000,000 credits (2,107,153)PPS Sieve Amethyst: Earned 1,000,000 credits (1,045,010)AP 26/27 Ruby: Earned 2,000,000 credits (2,470,273)WW Double Bronze: Earned 100,000,000 credits (130,620,000)GFN Turquoise: Earned 5,000,000 credits (7,149,778)PSA Gold: Earned 500,000 credits (678,219)
Message 145438 - Posted: 20 Nov 2020 | 18:09:38 UTC

I saw that LLR2 also allows numbers of form b^n - b^m - 1 as input. I never saw those before, what type is that?

Playing around with it a bit, I realized that they are actually just a special type of Riesel base b:

b^n - b^m - 1 = [b^(n-m) - 1] * b^m - 1


So why are they specifically mentioned?


____________
Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^8-1979 & 1281979 + 4 (cousin prime)

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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 Jade: Earned 10,000,000 credits (12,019,664)PSP LLR Silver: Earned 100,000 credits (212,242)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (145,419)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)WW Turquoise: Earned 5,000,000 credits (9,640,000)GFN Amethyst: Earned 1,000,000 credits (1,707,013)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 145457 - Posted: 20 Nov 2020 | 20:32:44 UTC
Last modified: 20 Nov 2020 | 20:35:25 UTC

How did you see it?

If I call LLR2 with -h, I see:

-q"expression" Test a single k*b^n+c or b^n-b^m+c number.


One special case is primes of form Phi(3, -(b^n)) where Phi(3, -x) is the third cyclotomic polynomial evaluated at negative x, which is x^2 - x + 1.

/JeppeSN

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Joined: 25 Feb 20
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321 LLR Amethyst: Earned 1,000,000 credits (1,058,073)Cullen LLR Amethyst: Earned 1,000,000 credits (1,169,946)ESP LLR Amethyst: Earned 1,000,000 credits (1,093,381)Generalized Cullen/Woodall LLR Amethyst: Earned 1,000,000 credits (1,148,593)PPS LLR Amethyst: Earned 1,000,000 credits (1,225,852)PSP LLR Amethyst: Earned 1,000,000 credits (1,248,861)SoB LLR Amethyst: Earned 1,000,000 credits (1,669,219)SR5 LLR Amethyst: Earned 1,000,000 credits (1,060,324)SGS LLR Amethyst: Earned 1,000,000 credits (1,108,160)TRP LLR Amethyst: Earned 1,000,000 credits (1,039,866)Woodall LLR Amethyst: Earned 1,000,000 credits (1,129,385)321 Sieve (suspended) Ruby: Earned 2,000,000 credits (2,107,153)PPS Sieve Amethyst: Earned 1,000,000 credits (1,045,010)AP 26/27 Ruby: Earned 2,000,000 credits (2,470,273)WW Double Bronze: Earned 100,000,000 credits (130,620,000)GFN Turquoise: Earned 5,000,000 credits (7,149,778)PSA Gold: Earned 500,000 credits (678,219)
Message 145487 - Posted: 21 Nov 2020 | 5:58:17 UTC - in response to Message 145457.
Last modified: 21 Nov 2020 | 5:59:49 UTC

b^n-b^m+c

For LLR2 c can be -1 or +1. I just looked at the -1 case, but +1 is the same. Then it's a special Proth number.

Of course it can also be any other number than -1/+1 but that it can still be expressed as k * b^n + c.

So why is it mentioned specifically?
____________
Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^8-1979 & 1281979 + 4 (cousin prime)

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Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 2 primes in the 2020 Tour de PrimesFound 2 primes in the 2021 Tour de Primes321 LLR Bronze: Earned 10,000 credits (38,954)ESP LLR Gold: Earned 500,000 credits (942,185)PPS LLR Double Silver: Earned 200,000,000 credits (284,953,062)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 (suspended) Silver: Earned 100,000 credits (395,205)PPS Sieve Double Silver: Earned 200,000,000 credits (229,814,554)TRP Sieve (suspended) Gold: Earned 500,000 credits (669,191)AP 26/27 Sapphire: Earned 20,000,000 credits (25,547,717)WW Double Amethyst: Earned 1,000,000,000 credits (1,289,832,000)GFN Emerald: Earned 50,000,000 credits (79,577,076)PSA Ruby: Earned 2,000,000 credits (3,025,234)
Message 145537 - Posted: 22 Nov 2020 | 2:42:38 UTC - in response to Message 145487.
Last modified: 22 Nov 2020 | 2:45:18 UTC

It actually would not be a Proth number. For a generalized Proth number k*b^n + 1 (or Thorp, k*b^n-1), the requirement is that k < b^n. For the rearrangement you provided, where k = b^(n-m)-1, it will not necessarily be true that k < b^m (such as when n-m > m)

Edit: Concerning your original question, programs usually have a limit on the size of k that are acceptable to test. With a different form, you don't have to fight with the k requirement.

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Joined: 25 Feb 20
Posts: 399
ID: 1241833
Credit: 157,022,093
RAC: 1,014,210
321 LLR Amethyst: Earned 1,000,000 credits (1,058,073)Cullen LLR Amethyst: Earned 1,000,000 credits (1,169,946)ESP LLR Amethyst: Earned 1,000,000 credits (1,093,381)Generalized Cullen/Woodall LLR Amethyst: Earned 1,000,000 credits (1,148,593)PPS LLR Amethyst: Earned 1,000,000 credits (1,225,852)PSP LLR Amethyst: Earned 1,000,000 credits (1,248,861)SoB LLR Amethyst: Earned 1,000,000 credits (1,669,219)SR5 LLR Amethyst: Earned 1,000,000 credits (1,060,324)SGS LLR Amethyst: Earned 1,000,000 credits (1,108,160)TRP LLR Amethyst: Earned 1,000,000 credits (1,039,866)Woodall LLR Amethyst: Earned 1,000,000 credits (1,129,385)321 Sieve (suspended) Ruby: Earned 2,000,000 credits (2,107,153)PPS Sieve Amethyst: Earned 1,000,000 credits (1,045,010)AP 26/27 Ruby: Earned 2,000,000 credits (2,470,273)WW Double Bronze: Earned 100,000,000 credits (130,620,000)GFN Turquoise: Earned 5,000,000 credits (7,149,778)PSA Gold: Earned 500,000 credits (678,219)
Message 145575 - Posted: 23 Nov 2020 | 18:10:27 UTC

It might not be called Proth, but it will be accepted by most software anyway, I think.

But if a very large k can be circumvented by writing it like this, maybe that's the reason.

I still think it's strange that it's specifically mentioned but apparently not really used (at least by active users here). I might ask at mersenneforum.
____________
Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^8-1979 & 1281979 + 4 (cousin prime)

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Message boards : General discussion : b^n - b^m - 1 primes

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