20190420, 01:01  #1 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{3}×7×163 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, p_{6} = p+6 ...and presto, done: 63166*3*2^534*(9*2^10681)+18*2^5341 is prime, quite trivially, and 63166*3*2^534*(9*2^10681)+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. 
20190420, 02:20  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10001110101000_{2} Posts 
Oneup
318725*3*2^2208*(9*2^44161)+18*2^22081 318725*3*2^2208*(9*2^44161)+18*2^2208+5 Next up 363629*3*2^3168*(9*2^63361)+18*2^31681 363629*3*2^3168*(9*2^63361)+18*2^3168+5 And slightly larger 2865046*7*2^6614*(49*2^132281)+42*2^66141 2865046*7*2^6614*(49*2^132281)+42*2^6614+5 
20190420, 03:44  #3 
Sep 2002
Database er0rr
2·1,723 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 20190420 at 03:46 
20190420, 03:56  #4 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{3}·7·163 Posts 
If you don't sieve, you will waste enormous amount of time on ineligible k values.
I sieved to 1T. 
20190420, 07:51  #5  
"Robert Gerbicz"
Oct 2005
Hungary
17×83 Posts 
Quote:
Let r=k*2^n+1 ~ sqrt(N) Proth prime, then search p in the form: Code:
p=c*r*2^n+6*r5 where c=1,2,3,.. is running. 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. 

20190420, 11:12  #6  
Sep 2002
Database er0rr
2·1,723 Posts 
Quote:
I have now written a sieve in parigp (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 

20190420, 19:43  #7 
Sep 2002
Database er0rr
2·1,723 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. 
20190420, 21:23  #8  
"Robert Gerbicz"
Oct 2005
Hungary
17×83 Posts 
Quote:
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. 

20190420, 21:40  #9 
"Robert Gerbicz"
Oct 2005
Hungary
17·83 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 0th coefficient of P: 0 Give the 1th coefficient of P: 18 Give the degree of the Q polynom: 2 Give the 0th coefficient of Q: 0 Give the 1th coefficient of Q: 1 Give the 2th coefficient of Q: 3 Give the number of c values for the sieve: 2 0th c value: 1 1th 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 #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 20190420 at 21:45 
20190420, 21:54  #10  
Sep 2002
Database er0rr
2·1,723 Posts 
Quote:
I downloaded and successfully compiled PolySieve.c. [code deleted due to cross post] Last fiddled with by paulunderwood on 20190420 at 21:58 

20190420, 22:31  #11  
"Robert Gerbicz"
Oct 2005
Hungary
17×83 Posts 
Quote:
Btw polysieve as you can see is single threaded and there is no save option, so don't stop it, however maybe from PuzzlePeter 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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