Double Wagstaff prime?

[ET_]

Originally Posted by lalera

hi,
mmff does not (yet) run with nvidia turing cards

It does, but under Linux.

[mathwiz]

Perhaps the author of mmff could comment on the feasibility of modifying this program to find Double Wagstaff factors, and if so, how it would need to be changed?

[GP2]

Originally Posted by mathwiz

Perhaps the author of mmff could comment on the feasibility of modifying this program to find Double Wagstaff factors, and if so, how it would need to be changed?

I'm not the author, but I'm looking at the source code. The code special-cases each of the exponents 31, 61, 89,107, 127 for Mersenne, plus a bunch of specific Fermat exponents. Wagstaff shares the exponents 31, 61, 127 with Mersenne, but also has the new exponents 43, 79, 101. So new assembler code would have to be written.

It is not nearly as simple as the changes that were made to mfaktc to handle Wagstaff, which was a mere handful of #ifdef statements. The mfaktc code can handle any arbitrary exponent up to a certain limit, including MM31 (exponent = about 2.1 billion), but not the huge "double" exponents. I used it to test WW31 (exponent = about 715 million) up to TF=2^80 and found one small factor of about 62 bits. WW43 (exponent = about 2.9 trillion) is out of reach.

The Double Mersenne search hasn't had any results beyond MM31. The most recent of four factors of MM31 was found in 2005, so probably mmff was not used to find it. Since Double Mersenne hasn't had much success and seems to have only two currently active users, and Wagstaff has much less overall interest than Mersenne, and creating Double Wagstaff code would involve non-trivial specialized effort, it's kind of hard to make a case for doing Double Wagstaff.

[Dylan14]

I don't know if there has been any update to searching for factors of double Wagstaffs, but I searched for potential factors of W(W(42737)), using pfgw64 with the -f1 flag, and the following input:

ABC2 2*$a*((2^42737+1)/3)+1
a: from 1 to 50000

this yielded the following PRP's and potential factors:

2*21085*((2^42737+1)/3)+1
2*42289*((2^42737+1)/3)+1

Unfortunately I don't have a program that can test for divisibility, so I can't test if they are factors or not.
Although this is reasonably quick (a test takes about half a second), there's a lot of candidates with a factor of 3, so a sieve would be good for a serious search.

[axn]

Wagstaff factors must be 1 or 3 (mod 8). Your numbers are 7 (mod 8). So they can't divide.

But if you want to test them, you can use Pari/GP

? W=(2^42737+1)/3;
? N=2*21085*((2^42737+1)/3)+1;
? Mod(2,N)^W+1 == 0
%1 = 0

EDIT:- Only search k= 0 or 3 (mod 4) for double wagstaff factors

[Dylan14]

Thanks axn for the advice. I've updated the ABC2 header:

ABC2 2*(4*$a)*((2^42737+1)/3)+1 | 2*(4*$a+3)*((2^42737+1)/3)+1
a: from 0 to 62500

The first expression tests k's which are 0 mod 4, and the second tests k's which are 3 mod 4. I got the following PRP's:

2*(4*13918+3)*((2^42737+1)/3)+1
2*(4*18327+3)*((2^42737+1)/3)+1
2*(4*26962+3)*((2^42737+1)/3)+1
2*(4*34111)*((2^42737+1)/3)+1 
2*(4*38520)*((2^42737+1)/3)+1 
2*(4*45025)*((2^42737+1)/3)+1 
2*(4*54945)*((2^42737+1)/3)+1 
2*(4*58207)*((2^42737+1)/3)+1 
2*(4*58308+3)*((2^42737+1)/3)+1

Unfortunately, none of these divide W(W(42737)).

[axn]

I did a search of all k < 10^6. Here are the primes. No factors, natch.

2*55675*((2^42737+1)/3)+1
2*73311*((2^42737+1)/3)+1
2*107851*((2^42737+1)/3)+1
2*136444*((2^42737+1)/3)+1
2*154080*((2^42737+1)/3)+1
2*180100*((2^42737+1)/3)+1
2*219780*((2^42737+1)/3)+1
2*232828*((2^42737+1)/3)+1
2*233235*((2^42737+1)/3)+1
2*266260*((2^42737+1)/3)+1
2*284271*((2^42737+1)/3)+1
2*308163*((2^42737+1)/3)+1
2*319195*((2^42737+1)/3)+1
2*340684*((2^42737+1)/3)+1
2*341500*((2^42737+1)/3)+1
2*348316*((2^42737+1)/3)+1
2*367168*((2^42737+1)/3)+1
2*368020*((2^42737+1)/3)+1
2*406035*((2^42737+1)/3)+1
2*427771*((2^42737+1)/3)+1
2*464971*((2^42737+1)/3)+1
2*468720*((2^42737+1)/3)+1
2*470836*((2^42737+1)/3)+1
2*534735*((2^42737+1)/3)+1
2*535951*((2^42737+1)/3)+1
2*574444*((2^42737+1)/3)+1
2*576928*((2^42737+1)/3)+1
2*580056*((2^42737+1)/3)+1
2*623875*((2^42737+1)/3)+1
2*628611*((2^42737+1)/3)+1
2*639783*((2^42737+1)/3)+1
2*652996*((2^42737+1)/3)+1
2*691695*((2^42737+1)/3)+1
2*759304*((2^42737+1)/3)+1
2*771864*((2^42737+1)/3)+1
2*795195*((2^42737+1)/3)+1
2*813655*((2^42737+1)/3)+1
2*823683*((2^42737+1)/3)+1
2*902104*((2^42737+1)/3)+1
2*925608*((2^42737+1)/3)+1
2*953796*((2^42737+1)/3)+1
2*995536*((2^42737+1)/3)+1