mersenneforum.org Let's find some large sexy prime pair (and, perhaps, a triplet)
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 2019-04-20, 01:01 #1 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 960810 Posts Let's find some large sexy prime pair (and, perhaps, a triplet) Let's start from Ken Davis' construction. Observe the form that he used (which is similar to the form J.K.Andersen used before him). Maybe we can find this even cheaper in computrons. Illustration: Let's take m=3*2^n, so that m+1 is prime. A tiny example m=3*2^534. Then we will sieve for two forms: p = k*m*(m^2 - 1)+ 6*m -1, p6 = p+6 ...and presto, done: 63166*3*2^534*(9*2^1068-1)+18*2^534-1 is prime, quite trivially, and 63166*3*2^534*(9*2^1068-1)+18*2^534+5 (with 3*2^534+1 as a helper) Now, repeat with m= 3*2^34350 3*2^42294 3*2^42665 3*2^44685 3*2^48150 3*2^55182 3*2^59973 The only part to write is a simple sieve, then sieve, and then do some PRP'ing.
 2019-04-20, 02:20 #2 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 23×1,201 Posts One-up 318725*3*2^2208*(9*2^4416-1)+18*2^2208-1 318725*3*2^2208*(9*2^4416-1)+18*2^2208+5 Next up 363629*3*2^3168*(9*2^6336-1)+18*2^3168-1 363629*3*2^3168*(9*2^6336-1)+18*2^3168+5 And slightly larger 2865046*7*2^6614*(49*2^13228-1)+42*2^6614-1 2865046*7*2^6614*(49*2^13228-1)+42*2^6614+5
 2019-04-20, 03:44 #3 paulunderwood     Sep 2002 Database er0rr 1111010110012 Posts I am working on 34350 using -f -o of pfgw for "-1" as feedback to pfgw -f later on an ABC file with +5 & -1 in the header... Last fiddled with by paulunderwood on 2019-04-20 at 03:46
 2019-04-20, 03:56 #4 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 960810 Posts If you don't sieve, you will waste enormous amount of time on ineligible k values. I sieved to 1T.
2019-04-20, 07:51   #5
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

5EE16 Posts

Quote:
 Originally Posted by Batalov Maybe we can find this even cheaper in computrons. Illustration: Let's take m=3*2^n, so that m+1 is prime. A tiny example m=3*2^534. Then we will sieve for two forms: p = k*m*(m^2 - 1)+ 6*m -1, p6 = p+6 ... Now, repeat with m= 3*2^34350 3*2^42294 3*2^42665 3*2^44685 3*2^48150 3*2^55182 3*2^59973
You can do it even much denser:
Let r=k*2^n+1 ~ sqrt(N) Proth prime, then search p in the form:

Code:
p=c*r*2^n+6*r-5 where c=1,2,3,.. is running.
Then p-1 is divisible by 2^n, and p+5 is divisible by r, what is a known prime.
Bingo, and the sieve is blazingly fast, because you need only 2^n mod s, where s is prime.

example:
Code:
k=165;n=100 for that r=k*2^n+1 is a Proth prime, and turned out that c=2920 is good.

2019-04-20, 11:12   #6
paulunderwood

Sep 2002
Database er0rr

3,929 Posts

Quote:
 Originally Posted by paulunderwood I am working on 34350 using -f -o of pfgw for "-1" as feedback to pfgw -f later on an ABC file with +5 & -1 in the header...
I did not get far.

I have now written a sieve in pari-gp (which I will convert to PrimeSieve+GMP) and am testing the world record contender:

p=c*(3*2^34350+1)*2^34350+6*(3*2^34350+1)-5

 2019-04-20, 19:43 #7 paulunderwood     Sep 2002 Database er0rr 3,929 Posts I just can't my sieves to behave Moreover, I feel that any record breaking sexy pair needs to involve primorials in order to get the required density when searching.
2019-04-20, 21:23   #8
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

27568 Posts

Quote:
 Originally Posted by paulunderwood I just can't my sieves to behave Moreover, I feel that any record breaking sexy pair needs to involve primorials in order to get the required density when searching.
That is just wrong assumption, if you'd be correct we would search only on the form say k*p#+1 and not Mersenne.

And for sieve why not use my ancient polysieve: https://primes.utm.edu/bios/page.php?id=3934 . That handle this problem also, I'll give how to feed this problem for the code.

 2019-04-20, 21:40 #9 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 2×3×11×23 Posts Here it is: Code: \$ ./polysieve Sieve P(s)+a*Q(s)+c for multiple c values, with fixed s=k*b^n+d; P,Q is polynom. Give k: 1 Give b: 2 Give n: 34350 Give d: 0 Give the degree of the P polynom: 1 Give the 0-th coefficient of P: 0 Give the 1-th coefficient of P: 18 Give the degree of the Q polynom: 2 Give the 0-th coefficient of Q: 0 Give the 1-th coefficient of Q: 1 Give the 2-th coefficient of Q: 3 Give the number of c values for the sieve: 2 0-th c value: 1 1-th c value: 7 Give start and end value for 'a' (in billions)! 0 10 Give the limit for sieving primes (maxp): 1000000000000 Give the name of the file to output the numbers! sexy.txt Using primes for wheelsieve up to 5 On line 18 of the c code set: #define bound_small_primes 5//11 // used up to this bound all primes in wheel sieve (change it, but it is very critical) to lower the wheelsieve. The range for 'a' is really what you like (what was c in my previous post), above we test 'a' from 0 to 10 (in billions), and maxp=1000000000000. Note that we needed to use the s=2^n to make it an integer polynom. ps. use smaller example to test it out, say 534 instead of the large 34350 (you need to change only that line) to handle that case. My code also works for the general case, on every Proth numbers (with some modification on the polynoms). Last fiddled with by R. Gerbicz on 2019-04-20 at 21:45
2019-04-20, 21:54   #10
paulunderwood

Sep 2002
Database er0rr

3,929 Posts

Quote:
 Originally Posted by R. Gerbicz That is just wrong assumption, if you'd be correct we would search only on the form say k*p#+1 and not Mersenne. And for sieve why not use my ancient polysieve: https://primes.utm.edu/bios/page.php?id=3934 . That handle this problem also, I'll give how to feed this problem for the code.
When searching for arithmetic progressions we have always used primorials -- I think this sexy types are similar, but I willing to give it another shot without them.

[code deleted due to cross post]

Last fiddled with by paulunderwood on 2019-04-20 at 21:58

2019-04-20, 22:31   #11
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

5EE16 Posts

Quote:
 Originally Posted by paulunderwood When searching for arithmetic progressions we have always used primorials -- I think this sexy types are similar, but I willing to give it another shot without them.
In general we use primorials for small numbers, say when you're searching 18 primes in ap, then the sieve bound is also smaller, and it is better to use primorials. In our case with primorials you'd only lost in sieving for these large numbers.

Btw polysieve as you can see is single threaded and there is no save option, so don't stop it, however maybe from Puzzle-Peter there is an updated code with save option somewhere in the forum. Not thought that somebody would run my code for weeks/months(?).

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