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 2022-08-15, 21:03 #1 CRGreathouse     Aug 2006 135438 Posts Software for finding prime constellations Suppose I have an admissible tuple (0, n_1, n_2, ..., n_{k-1}) and I am interested in finding instances of this constellation: numbers N such that N, N + n_1, ..., N + n_{k-1} are all prime. Is there software out there to do this search efficiently? In my immediate case, k = 19.
 2022-08-15, 21:05 #2 CRGreathouse     Aug 2006 135438 Posts https://arxiv.org/abs/1807.08777 is relevant, but I don't know of any implementations.
 2022-08-15, 21:27 #3 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 72×47 Posts 1st SM and now Mr. Greathouse. Are the stars lining up somewhere? My 2 cents: PFGW it’s the most efficient primality tester and you can script it so that it checks the 1st then and if and only if Prime would test next and so on. It inserts a noticeable line in the output when all items match (are Prime). It lacks an advanced sieving feature but you can (I have successfully, so should be a piece of cake for you) combine Pari-gp script and initiate pfgw when a sieved candidate is found. Obvious not telling you anything new, why would you look for another software? Nice to have you posting again. The forum had deteriorated without your inputs. Last fiddled with by a1call on 2022-08-15 at 21:59
 2022-08-16, 09:41 #4 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 2×19×31 Posts I don't know of a good software for searching for k-tuples I wrote some Maple code to find the smallest few k-tuples for k= 7,8,9,10,11,12,13. back in 2015 My Maple code and some prime k-tuples appear in OEIS.org Dana Jacobsen and Norman Luhn have extended these lists (and others) A link to this work is on my personal webpage https://mattanderson.fun/f/prime-constellations Also, Norman Luhn has a web page with lists of prime k-tuples, and the form that these numbers have. see - https://pzktupel.de/ktpatt_hl.php for example, form Norman's webpage, about 19-tuples, if the 19-tuple is {0 4 6 10 16 22 24 30 34 36 42 46 52 60 64 66 70 72 76} then the smallest prime in the constellation will have the form 37 (modulo 30030) I read this off the chart at pzktupel.de/ktpatt_hl.php There are 4 patterns for 19-tuples that are also prime constellations. They are shown on the chart. Currently, there is no known prime 19-tuple with pattern {0 4 6 10 12 16 24 30 34 40 42 46 52 54 60 66 70 72 76} Initial primes in this 19-tuple will have the form 29917 (modulo 30030) I pulled this information from this webpage https://pzktupel.de/SMArchiv/smadditions.html Maple code is slow, but easy for me to write code. I sped up the Maple isprime() function by first testing if a number has any factors less than 100, then, if all candidates of the k-tuple pass that test, run the full isprime() function on all 'k' of the possible primes. See my code for 10-tuples here http://oeis.org/A027569 Good luck on being the first to find this 19-tuple. I don't know of any downloadable software for finding admissable k-tuples. Norman Luhn has his software but it is not ready to be shared. Also, if someone can find a certain 447-tuple, they can show a counterexample to a conjecture of Hardy and Littlewood. see - http://www.opertech.com/primes/k-tuples.html Best of luck, and have a nice day.
2022-08-16, 11:56   #5
Dr Sardonicus

Feb 2017
Nowhere

32×677 Posts

Quote:
 Originally Posted by CRGreathouse Suppose I have an admissible tuple (0, n_1, n_2, ..., n_{k-1}) and I am interested in finding instances of this constellation: numbers N such that N, N + n_1, ..., N + n_{k-1} are all prime. Is there software out there to do this search efficiently? In my immediate case, k = 19.
If you go to cybertronic's pzktupel site, you'll find some 19-tuples (and some longer ones) listed, along with who found them and when. Perhaps he could point you in the right direction.

 2022-08-16, 19:11 #6 mart_r     Dec 2008 you know...around... 26×13 Posts AFAIK, Norman Luhn, Jens Kruse Andersen and others all use highly customized code. My code for the 13-tuples some years ago also was highly customized, written in Basic, and too slow for 19-tuplets. I haven't yet seen any general-purpose efficient software specifically for large k-tuples, but if there is one, I would also be interested
 2022-08-17, 18:45 #7 mart_r     Dec 2008 you know...around... 26·13 Posts Houston, we've got a program. And a problem. I've written a Pari program that captures the gist of the most important steps in a k-tuple finding algorithm. Problem is, it is quite slow. Finding 12- or 13-tuples can take several minutes, so it's most likely not suitable for 19-tuples. Code: print("ktuple(pattern as vector): tries to find tuples of primes with given pattern"); ktuple(p)= { h=2048; \\ lower bound in number of pre-wheel-sieved values - can be changed manually if so desired z=2^16; \\ trial division limit; tuples larger than z^2 get TD'd up to z, after that, numbers are checked sequentially if prime - can also be changed if necessary p=vecsort(p); \\ just in case pattern is not strictly increasing \\ check if pattern is admissible (a=1:yes, a=0:no) a=1; forprime(q=2,#p, v=vector(q); for(i=1,#p, v[q-p[i]%q]=1 ); v=vecsort(v); if(v[1]==1, print("pattern is not admissible / all residues mod "q" are covered"); a=0; break() ) ); \\ find starting point s and difference pattern d; wheel of length q# for first q such that #d > h if(a, s=p[1]%2+1; d=[2]; q=1; m=2; while(#d1, print1("s") ); print(" in wheel of length "m) ); print(); \\ residue patterns in vectors w=p[#p]-p[1]; o=vector(w); for(i=1,#p-1, o[p[#p]-p[i]]=1 ); l=primepi(w)-primepi(q); q=nextprime(q+2); if(l>0, g=vector(w); n=vector(l); k=0; forprime(c=q,w, k++; g[c]=k; n[k]=vector(c); for(i=1,#p-1, j=(p[#p]-p[i])%c; if(j, n[k][j]=1 ) ) ) ); s+=p[#p]; f=0; print("searching for primes with {pattern} = "p); \\ main search while(s p=[0,2,6,12,20,30,42,56,72,90,110] %50 = [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110] (19:55) gp > ktuple(p) 1 admissible pattern in wheel of length 6 2 admissible patterns in wheel of length 30 6 admissible patterns in wheel of length 210 30 admissible patterns in wheel of length 2310 180 admissible patterns in wheel of length 30030 1440 admissible patterns in wheel of length 510510 12960 admissible patterns in wheel of length 9699690 searching for primes with {pattern} = [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110] 180078317 + {pattern} 1278189947 + {pattern} 1761702947 + {pattern} 1829187287 + {pattern} 5862143447 + {pattern} Last fiddled with by mart_r on 2022-08-17 at 18:57 Reason: removed a redundant semicolon
2022-08-18, 04:08   #8
LaurV
Romulan Interpreter

"name field"
Jun 2011
Thailand

240058 Posts

Quote:
 Originally Posted by CRGreathouse Suppose I have ..
Welcome back!

2022-08-18, 14:57   #9
science_man_88

"Forget I exist"
Jul 2009
Dartmouth NS

841810 Posts

Quote:
 Originally Posted by mart_r Code: print("ktuple(pattern as vector): tries to find tuples of primes with given pattern"); ktuple(p)= { h=2048; \\ lower bound in number of pre-wheel-sieved values - can be changed manually if so desired z=2^16; \\ trial division limit; tuples larger than z^2 get TD'd up to z, after that, numbers are checked sequentially if prime - can also be changed if necessary p=vecsort(p); \\ just in case pattern is not strictly increasing \\ check if pattern is admissible (a=1:yes, a=0:no) a=1; forprime(q=2,#p, v=vector(q); for(i=1,#p, v[q-p[i]%q]=1 ); v=vecsort(v); if(v[1]==1, print("pattern is not admissible / all residues mod "q" are covered"); a=0; break() ) ); \\ find starting point s and difference pattern d; wheel of length q# for first q such that #d > h if(a, s=p[1]%2+1; d=[2]; q=1; m=2; while(#d1, print1("s") ); print(" in wheel of length "m) ); print(); \\ residue patterns in vectors w=p[#p]-p[1]; o=vector(w); for(i=1,#p-1, o[p[#p]-p[i]]=1 ); l=primepi(w)-primepi(q); q=nextprime(q+2); if(l>0, g=vector(w); n=vector(l); k=0; forprime(c=q,w, k++; g[c]=k; n[k]=vector(c); for(i=1,#p-1, j=(p[#p]-p[i])%c; if(j, n[k][j]=1 ) ) ) ); s+=p[#p]; f=0; print("searching for primes with {pattern} = "p); \\ main search while(s p=[0,2,6,12,20,30,42,56,72,90,110] %50 = [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110] (19:55) gp > ktuple(p) 1 admissible pattern in wheel of length 6 2 admissible patterns in wheel of length 30 6 admissible patterns in wheel of length 210 30 admissible patterns in wheel of length 2310 180 admissible patterns in wheel of length 30030 1440 admissible patterns in wheel of length 510510 12960 admissible patterns in wheel of length 9699690 searching for primes with {pattern} = [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110] 180078317 + {pattern} 1278189947 + {pattern} 1761702947 + {pattern} 1829187287 + {pattern} 5862143447 + {pattern}
Looks like it should be highly paralleled.

2022-08-18, 17:14   #10
mart_r

Dec 2008
you know...around...

26·13 Posts

Quote:
 Originally Posted by science_man_88 Looks like it should be highly paralleled.
Thanks for the reply! You make me happy

Different patterns can be looked at in parallel, but manually setting a starting point would also be nice. Should be no problem for me to implement.
I have a couple more ideas for improvements. But I also have to earn some money, so give me a day or three...

 2022-08-21, 18:42 #11 mart_r     Dec 2008 you know...around... 34016 Posts Just practicing my programming skills It would still take thousands of years to find a prime novemdecuplet with my program, but it might be useful for small searches and doublechecks. - Working with "concat" was slowing the wheel sieve down. Fixed that, it's now way faster. - The search itself is now also about 20% faster (for medium-sized tuples). And since the wheel can now be made larger, another significant speedup is possible for large tuples. - A starting point can be set. Example: "ktuple([1,3,7,9],10^49)" finds the first quadruplets with 50 digits. - (Did anyone notice that the first member of the input pattern need not be zero? You can also search for quintuplets with pattern [-4,0,2,6,8], for instance.) - Optional write to file and optional output tracking. I've finally learned how to use the input command properly. - There's a lot of room for optimisation by experimenting with different wheel sieve sizes and trial division depths. This is left as an exercise for the user ;) - Any errors or bugs? Feel free to report. Code: ktuple(p,x)= { h=8192; \\ lower bound in number of pre-wheel-sieved values - can be changed manually if so desired z=1024*#p; \\ trial division limit - can also be changed if necessary print1("Write output file "Strchr(34)"ktuple_output.txt"Strchr(34)"? 1:yes / 0:no ");l=input(); print1("Output search status every ___ seconds? (0/default: don't output search status, only tuples) ");j=input(); p=vecsort(p); \\ just in case pattern is not strictly increasing \\ check if pattern is admissible (a=1:yes, a=0:no) a=1; y=vector(#p); forprime(q=2,#p, v=vector(q); for(i=1,#p, y[q]+=1-v[q-p[i]%q]; v[q-p[i]%q]=1 ); v=vecsort(v); if(v[1]==1, print("pattern is not admissible / all residues mod "q" are covered"); a=0; break() ) ); \\ find admissible initial members s[f]; generate wheel of size q# := m for first q such that d := #s[f] >= h if(a, s=[p[1]%2+1]; d=1; q=1; m=2; while(d#p, c=0; v=vector(q); for(i=1,#p, c+=1-v[q-p[i]%q]; v[q-p[i]%q]=1 ) , c=y[q] ); d*=(q-c); t=vector(d); f=0; u=0; e=0; b=1; while(e#s, f=1; u+=m ); for(i=1,#p, b*=sign((u+s[f]+p[i])%q) ); if(b, e++; t[e]=u+s[f] , b=1 ) ); s=t; m*=q; print1(d" admissible pattern"); if(d>1, print1("s") ); print(" in wheel of size "m) ); print(); \\ residue patterns in vectors w=p[#p]-p[1]; o=vector(w); for(i=1,#p-1, o[p[#p]-p[i]]=1 ); f=primepi(w)-primepi(q); q=nextprime(q+2); if(f>0, g=vector(w); n=vector(f); k=0; forprime(c=q,w, k++; g[c]=k; n[k]=vector(c); for(i=1,#p-1, e=(p[#p]-p[i])%c; if(e, n[k][e]=1 ) ) ) ); u=(x\m)*m+p[#p]; if(u==p[#p], print("Patterns with initial member <= "precprime(q-2)" are not shown in the output!"); print(); if(l, write("ktuple_output.txt",Str("Patterns with initial member <= "precprime(q-2)" are not shown in the output!")) ) ); print("Searching for primes with {pattern} = "p); if(l, write("ktuple_output.txt",Str("Searching for primes with {pattern} = "p)) ); \\ main search (stage 1: complete trial division) j=(j*1000\256)*256; t=0; gettime(); while(uj, t=0; print("searching ... "u-p[#p]) ) ) ); \\ main search (stage 2: trial division up to z, then PRP test) while(1, for(f=1,d, x=u+s[f]; b=1; forprime(c=q,w, r=x%c; if(r==0, b=0; break() , if(n[g[c]][r], b=0; break() ) ) ); if(b, forprime(c=w+1,z, r=x%c; if(r==0, b=0; break() , if(r<=w, if(o[r], b=0; break() ) ) ) ); if(b, for(i=1,#p, if(!ispseudoprime(x-p[#p]+p[i]), b=0; break() ) ); if(b, print(x-p[#p]" + {pattern}"); if(l, write("ktuple_output.txt",Str(x-p[#p]" + {pattern}")) ) ) ) ) ); u+=m; if(j, t+=gettime(); if(t>j, t=0; print("searching ... "u-p[#p]) ) ) ) ) }; print("ktuple(pattern as vector [, starting point]): tries to find tuples of primes with given pattern [optionally start at x]")

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