Riesel Primes k*2^n-1, k<300 [Was "k=1"]

[Kosmaj]

Thanks for your helpful suggestion. I'm not aware that any available sieving software is checking for algebraic factors. What are you saying is that if k*2^n is a d-th power of b then k*2^n-1 is divisible by the b, which is obvious enough but we never paid attention to it.

BTW, k=125 is also affected because we are looking for n's divisible by 3 not by 5, not so?
The status of affected k<300 is this.
k=9, all even n's should be removed, checked to n=828k, currently not tested by anobody.
k=27, all n's divisible by 3 should be removed, currently being tested by the 12121 project. I'll mail Justin to tell him.
k=81, all even n's should be removed, currently being tested by Flatlander. I'll inform him.
k=125, all n's divisible by 3 should be removed, currently being tested by Flatlander. I'll inform him.
k=225, all even n's should be removed, checked to n=700k, currently being tested by Masser (but not actively?), I'll inform him.
k=243, all n's divisible by 5 should be removed, checked to n=260k and some larger n's, I have no information whether it's being tested or not.

BTW, I did some quick tests using ksieve and I found that the percentage of above "obvious" n's is small, not larger than 2-5% of all n's. For example in first 3 available files of the 27*2^n-1 project (beginning at n=851110) among 120 candidates there are 4 divisible by 3, or 3.33%. The file has been sieved to 1.7T. Here is an example
(27*2^851550-1)%(3*2^283850-1) = 0.

Thanks again!

Edit: k=81 is all right when viewed as 3^4 (for n=4*s, 81*2^n-1 is divisible by 5) but affected when expressed as 9^2.

[R. Gerbicz]

Originally Posted by Kosmaj

I'm not aware that any available sieving software is checking for algebraic factors.

There is such software: Multisieve for various type of numbers: cullen/woodall etc. But it isn't sieving k*2^n-1.

Originally Posted by Kosmaj

BTW, k=125 is also affected because we are looking for n's divisible by 3 not by 5, not so?

Yes, k=125 is also affected. It was a mistake in my previous post.

[Kosmaj]

Yes, I'm aware that Multisieve can do it, and I'm actively using it for generalized Cullen/Woodall. BTW, can you think of more algabraic factors (of other type) of k*2^n-1 for some specifc k/n values with or without the k<300 restriction? Thanks.

[R. Gerbicz]

Originally Posted by Kosmaj

BTW, can you think of more algabraic factors (of other type) of k*2^n-1 for some specifc k/n values with or without the k<300 restriction?

There is no other type of algebraic factors for k*2^n-1.

[Flatlander]

Hi all.

I am still testing k=81 and k=125.
k= 81 is now tested n>476000. (Sieved to n=500,000.)
k= 125 is now tested n>581000. (Sieved to n=1000,000.)
No new primes since the ones already reported some weeks ago.

I already have some BASIC code for removing candidates from sieved files.
To make sure I understand:

K=125
Remove all candidates where (n mod 3) = 0.

k=81
Remove all candidates where (n mod 2) = 0.

I will keep all rejected candidates in a seperate file and double check them by hand.

[R. Gerbicz]

Originally Posted by Flatlander

To make sure I understand:
K=125 Remove all candidates where (n mod 3) = 0.
k=81 Remove all candidates where (n mod 2) = 0.

Yes, those k*2^n-1 numbers has got algebraic factors.

[Flatlander]

Statistics.

k=81:
1.81% of remaining candidates were removed.

k=125:
3.96% of remaining candidates were removed.

A nice little speed boost. Thanks guys!

[Thomas11]

k=155 has been done to n=1M and I'm stopping there.
Unfortunately I haven't found any prime in the n=0.5-1M range

[Flatlander]

Tested to n=500,000. No more primes.

Releasing this k.

[sjtjung]

status report

k=103 is complete to 356k and continuing.

-steven

[Kosmaj]

Thanks for the report. You are now in a region which most likely was never checked before, and I think you have a good chance for a prime.

Happy hunting!