Let's find some large sexy prime pair (and, perhaps, a triplet)

[Batalov]

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.

[Batalov]

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

[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...

[Batalov]

If you don't sieve, you will waste enormous amount of time on ineligible k values.
I sieved to 1T.

[R. Gerbicz]

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:

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:

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

[paulunderwood]

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

[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.

[R. Gerbicz]

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.

[R. Gerbicz]

Here it is:

$ ./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).

[paulunderwood]

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.

I downloaded and successfully compiled PolySieve.c.

[code deleted due to cross post]

[R. Gerbicz]

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(?).