Cullen prime problem

Citrix · 2006-11-24, 21:03

On http://primes.utm.edu/top20/page.php?id=6
There is an open problem to find a cullen prime p*2^p+1 where p is also prime. Has anyone done anywork on this?

I tried to write a crude sieving program. There are ~9000 candidates left under p=5 million when the range is sieved to 350 Million.

Factors available on request.

Is anyone interested in improving on these results. Writing a faster sieve software? Any thoughts on how to solve this problem?

Thanks!

T.Rex · 2006-11-24, 21:15

There is a mistake on this page.
Cullen numbers are: n*2^n+1 and the last one is said to be: 338707 · 2^1354830+1 (a*2^b+1 where a!=b !!!!).
On the page:Cullen Primes it is said that the new and biggest one is: 1354828*2^1354828+1 .
T.

smh · 2006-11-24, 21:30

Quote:

Originally Posted by T.Rex

There is a mistake on this page.
Cullen numbers are: n*2^n+1 and the last one is said to be: 338707 · 2^1354830+1 (a*2^b+1 where a!=b !!!!).
On the page:Cullen Primes it is said that the new and biggest one is: 1354828*2^1354828+1 .
T.

338707*2^1354830+1 = 1354828*2^1354828+1

T.Rex · 2006-11-24, 21:37

Quote:

Originally Posted by smh

338707*2^1354830+1 = 1354828*2^1354828+1

Oh, yes!
But what a strange idea to write the number in a way different than his definition !
338707 is prime and 1354828 is the first even n satisfying n*2^n+1 .
T.

Citrix · 2006-11-24, 22:57

Quote:

Originally Posted by T.Rex

Oh, yes!
But what a strange idea to write the number in a way different than his definition !
338707 is prime and 1354828 is the first even n satisfying n*2^n+1 .
T.

The challenge is to find p*2^p+1 so this does not qualify as a solution.

rogue · 2006-11-25, 01:27

Quote:

Originally Posted by Citrix

On http://primes.utm.edu/top20/page.php?id=6
There is an open problem to find a cullen prime p*2^p+1 where p is also prime. Has anyone done anywork on this?

I tried to write a crude sieving program. There are ~9000 candidates left under p=5 million when the range is sieved to 350 Million.

Factors available on request.

Is anyone interested in improving on these results. Writing a faster sieve software? Any thoughts on how to solve this problem?

Thanks!

d/l my MultiSieve program. You can build a list of primes into an ABC file (which MultiSieve supports), then start sieving at 2.

Citrix · 2006-11-25, 05:55

Quote:

Originally Posted by rogue

d/l my MultiSieve program. You can build a list of primes into an ABC file (which MultiSieve supports), then start sieving at 2.

Your program is the first thing I tried, but your program is about 10 times slower than my crude program. (A range of 1 million with n= 0 to 5 M, takes 15 sec to sieve on a intel celeron 1.5 ghz, for my program) So I think your program is not well suited for this kind of search. What do you think?

Both the programs use the same algorithm, as much as I know. Since the gap between 2 numbers remaining can be as large as 1500. I think your program, has to compute that in steps of 50 etc..., this is where it slows down. There is no way, my coding skills can be better than yours.

I have attached the abc file, try analyzing it for speed ups. Attached file is sieved upto 150 Million n=5 million. If your program finds factors under 150M, please let me know, so I can debug my program.

Thanks!

rogue · 2006-11-25, 19:58

Quote:

Originally Posted by Citrix

Your program is the first thing I tried, but your program is about 10 times slower than my crude program. (A range of 1 million with n= 0 to 5 M, takes 15 sec to sieve on a intel celeron 1.5 ghz, for my program) So I think your program is not well suited for this kind of search. What do you think?

Both the programs use the same algorithm, as much as I know. Since the gap between 2 numbers remaining can be as large as 1500. I think your program, has to compute that in steps of 50 etc..., this is where it slows down. There is no way, my coding skills can be better than yours.

10 times? That almost sounds like an exaggeration. I suppose it is possible though. Remember, MultiSieve supports multiple bases and any n, not just where n is a prime. Since you are using only base 2, there might be base 2 tricks that I'm not using. If you send me your source I can compare our code and see if there are any tricks I can take advantage of. It has been a couple years since I've worked on the code and I have learned a few things since then, but probable not enough to account for such a huge gap. It doesn't help that I don't have any code optimization tools on x86.

Citrix · 2006-11-28, 22:57

For small ranges like people at prothsearch.net work on, my program is slightly slower than yours, but when it comes to just looking cullens where the index is prime, my program becomes several times faster. I assumed this was because, your program calculated the gap in increments of 50 ... etc.

Here is the pseudo code, if you still want the real source code, it will take a while to clean it up, before I email it to you.

- generate a prime p
-calculate the maximum gap between 2 candidates
-Create array and store all values 2^1, 2^2 ... 2^max_gap (All mod p), this is done by doing *2 mod p.
- Find the largest cullen left unsieved=max. Calculated 2^(-1*max)
- Go down from the 2^(-1*max) and multiply it by gaps, consecutively and check if this value equals the index of the cullen number being tested.

I assume your program uses this algorithm. For multiplication my program uses 64 bit ints, so program is limited to 32 bits. ( haven't implemented any of the special multiplication speeds up.)

Program is compiled under, VC++ 6.0 with some optomizations selected. Not sure if that makes any difference.

What do you think?

maxal · 2006-12-01, 20:48

There is a faster way to sieve out numbers of the form q*2^q+1 that are divisible by a given prime p>=3.

Suppose that q*2^q+1 is divisible by p. Then q*2^q = -1 (mod p) or alternatively q = - 2^(-q) (mod p).

Let q = -k (mod p-1) where 0 <= k < p-1, then q = - 2^(-q) = - 2^k (mod p).
From q = -k (mod p-1) and q = - 2^k (mod p) we can reconstruct q modulo p*(p-1) using Chinese theorem:
q = -k * p + 2^k * (p-1) (mod p*(p-1)) .............. (x)

In other words, for every k=0,...,p-2, there is an unique solution modulo p*(p-1) to the system of congruences
q*2^q + 1 = 0 (mod p)
q = -k (mod p-1)
and this solution is given by formula (x).

Therefore, sieving can be done as follows: for each prime p in selected range, let k go from 0 to p-2, compute
r = ((p-1-k) * p + (2^k mod p) * (p-1)) mod (p*(p-1)) (i.e., the smallest non-negative solution to (x)), and eliminate candidates
r, r + (p*(p-1)), r + 2*(p*(p-1)), ..., up to selected upper bound (for large p, it may appear that r is greater than the upper bound, in which case no actual elimination happens).

P.S. Of course, the powers (2^k mod p) can be computed incrementally as k goes from 0 to p-2:
having 2^k mod p=y computed, the value of 2^(k+1) mod p is computed as (y*2) mod p.

P.P.S. There is another speed up possible if we replace p-1 with the multiplicative order of 2 modulo p (that is a divisor of p-1). So, instead of letting k go from 0 to p-2, we compute incrementally and store values u[i]=2^i mod p, i=1,2,... until we get u[z]=1, meaning that z is the multiplicative order of 2 modulo p. Then we let k go from 0 to z-1, compute
r = ((z-k) * p + u[k] * z) mod (p*z), and eliminate candidates r, r + (p*z), r + 2*(p*z), ..., up to selected upper bound.

rogue · 2006-12-01, 21:42

I've looked at your post, i.e. Citrix's post, and see that he does use the same logic. I understand now why his code is more efficient. It is as he has stated. My code is tuned to sieving small ranges of consecutive n, which are fairly dense even after deep sieving. Since Citrix is using a large range of n where n must be prime, the density is much less (probably by a factor of between 20 and 30). The difference between our code would be even greater if his range were larger, but would be less if his range were smaller.

I could probably modify my code to use a dynamicly sized table (instead of a staticly sized table) to hold the values from 2^1 to 2^maxdiff (mod p). I suspect it would be much closer (and possibly faster) than Citrix's code. Citrix, if you are interested I could send you the source and point you to where the change needs to be made.

Maxal, I'll have to chew on your post a while. I think I have seen that suggestion before, but I didn't pursue it because as p gets large WRT the size of the range of n, it becomes much slower.

2006-11-24, 21:03	#1
Citrix Jun 2003 11001010101₂ Posts	Cullen prime problem (How everything began) On http://primes.utm.edu/top20/page.php?id=6 There is an open problem to find a cullen prime p2^p+1 where p is also prime. Has anyone done anywork on this? I tried to write a crude sieving program. There are ~9000 candidates left under p=5 million when the range is sieved to 350 Million. Factors available on request. Is anyone interested in improving on these results. Writing a faster sieve software? Any thoughts on how to solve this problem? Thanks! Last fiddled with by hhh on 2007-03-24 at 23:21*

2006-12-01, 20:48	#10
maxal Feb 2005 100000100₂ Posts	There is a faster way to sieve out numbers of the form q2^q+1 that are divisible by a given prime p>=3. Suppose that q2^q+1 is divisible by p. Then q2^q = -1 (mod p) or alternatively q = - 2^(-q) (mod p). Let q = -k (mod p-1) where 0 <= k < p-1, then q = - 2^(-q) = - 2^k (mod p). From q = -k (mod p-1) and q = - 2^k (mod p) we can reconstruct q modulo p(p-1) using Chinese theorem: q = -k * p + 2^k * (p-1) (mod p(p-1)) .............. (x) In other words, for every k=0,...,p-2, there is an unique solution modulo p(p-1) to the system of congruences q2^q + 1 = 0 (mod p) q = -k (mod p-1) and this solution is given by formula (x). Therefore, sieving can be done as follows: for each prime p in selected range, let k go from 0 to p-2, compute r = ((p-1-k) p + (2^k mod p) * (p-1)) mod (p(p-1)) (i.e., the smallest non-negative solution to (x)), and eliminate candidates r, r + (p(p-1)), r + 2(p(p-1)), ..., up to selected upper bound (for large p, it may appear that r is greater than the upper bound, in which case no actual elimination happens). P.S. Of course, the powers (2^k mod p) can be computed incrementally as k goes from 0 to p-2: having 2^k mod p=y computed, the value of 2^(k+1) mod p is computed as (y2) mod p. P.P.S. There is another speed up possible if we replace p-1 with the multiplicative order of 2 modulo p (that is a divisor of p-1). So, instead of letting k go from 0 to p-2, we compute incrementally and store values u[i]=2^i mod p, i=1,2,... until we get u[z]=1, meaning that z is the multiplicative order of 2 modulo p. Then we let k go from 0 to z-1, compute r = ((z-k) p + u[k] * z) mod (pz), and eliminate candidates r, r + (pz), r + 2(pz), ..., up to selected upper bound. Last fiddled with by maxal on 2006-12-01 at 21:11

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2006-11-24, 21:15	#2
T.Rex Feb 2004 France 2·7·67 Posts	There is a mistake on this page. Cullen numbers are: n2^n+1 and the last one is said to be: 338707 · 2^1354830+1 (a2^b+1 where a!=b !!!!). On the page:Cullen Primes it is said that the new and biggest one is: 1354828*2^1354828+1 . T.

2006-11-28, 22:57	#9
Citrix Jun 2003 3125₈ Posts	For small ranges like people at prothsearch.net work on, my program is slightly slower than yours, but when it comes to just looking cullens where the index is prime, my program becomes several times faster. I assumed this was because, your program calculated the gap in increments of 50 ... etc. Here is the pseudo code, if you still want the real source code, it will take a while to clean it up, before I email it to you. - generate a prime p -calculate the maximum gap between 2 candidates -Create array and store all values 2^1, 2^2 ... 2^max_gap (All mod p), this is done by doing 2 mod p. - Find the largest cullen left unsieved=max. Calculated 2^(-1max) - Go down from the 2^(-1*max) and multiply it by gaps, consecutively and check if this value equals the index of the cullen number being tested. I assume your program uses this algorithm. For multiplication my program uses 64 bit ints, so program is limited to 32 bits. ( haven't implemented any of the special multiplication speeds up.) Program is compiled under, VC++ 6.0 with some optomizations selected. Not sure if that makes any difference. What do you think?

2006-12-01, 21:42	#11
rogue "Mark" Apr 2003 Between here and the 7190₁₀ Posts	I've looked at your post, i.e. Citrix's post, and see that he does use the same logic. I understand now why his code is more efficient. It is as he has stated. My code is tuned to sieving small ranges of consecutive n, which are fairly dense even after deep sieving. Since Citrix is using a large range of n where n must be prime, the density is much less (probably by a factor of between 20 and 30). The difference between our code would be even greater if his range were larger, but would be less if his range were smaller. I could probably modify my code to use a dynamicly sized table (instead of a staticly sized table) to hold the values from 2^1 to 2^maxdiff (mod p). I suspect it would be much closer (and possibly faster) than Citrix's code. Citrix, if you are interested I could send you the source and point you to where the change needs to be made. Maxal, I'll have to chew on your post a while. I think I have seen that suggestion before, but I didn't pursue it because as p gets large WRT the size of the range of n, it becomes much slower.