20121203, 16:56  #1 
Sep 2006
The Netherlands
2×17×23 Posts 
Speeding up 321
For the search 3 * 2^n  1
I noticed the formula some similarity to Mersenne just of course that factor 3. So maybe there is some additional logics possible. 3 * 2^n  1 = 2^(n+1) + 2^n  1 So in fact our 'n' should be 2x larger. Maybe idea is speedup search by just checking all exponents n which holds true that 2n + 1 = prime I checked that for the big exponents a number of them indeed is prime. 3 out of 4 i checked. Didn't check rest. So it's a heuristic, not absolute truth. Then we also initially just search for numbers n that are multiple of 5, though this is much weaker heuristic so maybe ignore that one at later stage. Maybe can find quickly some prime numbers. Idea for primegrid to filter those out and do those first? 
20121203, 17:22  #2  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:
3*2^n1 is not 3*(2^n1) , please follow PEDMAS , all n>0 create values that are 5 mod 6 . 

20121203, 17:42  #3  
Sep 2006
The Netherlands
30E_{16} Posts 
Quote:
2n + 1 is prime for 3 out of 4 with n being: 2312734, 3136255, 4235414, 6090515 The idea is to speedup prime search by using heuristics  producing heuristics my expertise yet in math is bit more complicated thanks to all the math rules it follows usually :) What's odds that if you pick 4 random numbers of 7 digits that 3 out of 4 of them are prime for 2n+1 ? Last fiddled with by diep on 20121203 at 17:45 

20121203, 18:08  #4  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
Last fiddled with by science_man_88 on 20121203 at 18:11 

20121203, 18:17  #5  
Sep 2006
The Netherlands
1100001110_{2} Posts 
Quote:
Go take your medicine and redo your math with 7 or 8 digit numbers please. 

20121203, 18:22  #6 
"Forget I exist"
Jul 2009
Dumbassville
20C0_{16} Posts 

20121203, 20:36  #7 
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
2^{5}·11·17 Posts 
I wonder whether it would be possible to make a primality test for all Solinas primes.
Obviously is in reality just . I just worked out that This means that all Solinas primes are Riesel/Sierpinski primes. Solinas primes can all be proven prime by the same means. If then the llr and proth tests won't work. A N+1 or N1 test can be used. might need to be partially factored. edit: In fact this can be generalized further to: and I wonder if a primality test is possible for any Last fiddled with by henryzz on 20121203 at 21:34 
20121204, 01:07  #8  
Sep 2006
The Netherlands
2·17·23 Posts 
Quote:
Note i wrote the expression in 2^(n+1) + 2^n  1 in order to get to the 2n + 1 as heuristic to dig up some of the gold quicker, no other reasons. 321 is binary 0b101111...111111 so it's pretty trivial that if we take the 2 top bits which is 10 that this represents 31 so that's 3 * 2^n 1 As we are just 1 bitflip away from having a mersenne here, and Mersenne has a fast way of trial factoring and also for its exponents only prime numbers apply, i wonder which clever constraints are there for 321 formula. Now several Paul Underwoods, from which i met a few, have been sitting there in cabins at several locations in the UK pondering about this for years, so maybe there is no clever trick out there unlike with Mersenne. Of course the dark december months are a good excuse to reevaluate all this again :) Last fiddled with by diep on 20121204 at 01:21 

20121204, 15:32  #9 
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
1760_{16} Posts 
Should have spotted this last night.
Assuming is small/factorable enough it is now in N1 or N+1 test form. 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Speeding up (n mod m) for very large fixed n and varying smaller m  mickfrancis  Computer Science & Computational Number Theory  9  20160329 19:00 
Speeding up Mersenne hunting  m_f_h  Math  8  20070518 13:49 
Speeding up double checking when first test returns "prime"  Unregistered  PrimeNet  16  20060228 02:00 
Speeding up ECM 10x  clowns789  Software  2  20040811 03:45 