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

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Bur
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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?

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1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 4,400,000

JeppeSN

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Message 145457 - Posted: 20 Nov 2020 | 20:32:44 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

Bur
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Message 145487 - Posted: 21 Nov 2020 | 5:58:17 UTC - in response to Message 145457.

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?
____________
1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 4,400,000

Gelly
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Message 145537 - Posted: 22 Nov 2020 | 2:42:38 UTC - in response to Message 145487.

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.

Bur
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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.
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1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 4,400,000

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