20190719, 04:04  #254  
Jun 2010
191 Posts 
Quote:
Code:
Range Smallest First Twin k nvalue 10001999 177 1032 20002999 4359 2191 30003999 1149 3283 40004999 2565 4901 50005999 5775 5907 60006999 4737 6634 70007999 33957 7768 80008999 459 8529 90009999 33891 9869 1000010999 10941 10601 1100011999 915 11455 1200012999 73005 12178 1300013999 3981 13153 1400014999 175161 14171 1500015999 74193 15770 1600016999 138153 16436 1700017999 14439 17527 1800018999 56361 18989 1900019999 53889 19817 2000020999 7485 20023 2100021999 195045 21432 2200022999 31257 22312 2300023999 396213 23672 2400024999 177141 24365 2500025999 577065 25879 2600026999 182697 26172 2700027999 70497 27652 2800028999 445569 28353 2900029999 815751 29705 3000030999 249435 30977 3100031999 440685 31989 3200032999 51315 32430 3300033999 143835 33826 3400034999 959715 34895 3500035999 338205 35351 3600036999 47553 36172 3700037999 201843 37630 3800038999 683145 38746 3900039999 126423 39606 4000040999 604329 40315 4100041999 358965 41653 4200042999 272139 42379 4300043999 441201 43167 4400044999 >1M ??? 4500045999 311541 45439 4600046999 >1M ??? 4700047999 103893 47122 4800048999 694599 48501 4900049999 197109 49733 

20200609, 22:09  #255 
"Sam"
Nov 2016
311 Posts 
k*b^n+1 with k < n?
Also another interesting problem if anyone's interested:
Twin primes of the form k*b^n+1 with k < n > Due to the limited choices of fixing only base b, there are extremely rare. I tested some bases (3, 5, 6, 7, 10, 11, 12). Here are the largest twins found to n=2K (except b=3, which is checked to n=10K). Quite small, I tell you: Second twin (p+2): Code:
2618*3^4286+1 336*5^765+1 613*6^1922+1 525*10^632+1 1182*11^1409+1 860*12^967+1 Last fiddled with by carpetpool on 20200609 at 22:10 
20200610, 21:15  #256  
"Dylan"
Mar 2017
2×257 Posts 
Quote:
Code:
#script to automate sieving for small twins of the form k*b^n+/1, where #k < n import subprocess #set parameters b = input("Enter a base (not a perfect power):") minn = input("Enter the minimum n to test:") maxn = input("Enter the maximum n to test:") #check that minn is not 1. Otherwise the only k we would test is k = 0, but #0*b^1+/1 = +/1 for all b. And +1 and 1 are not prime (by definition). if int(minn) == 1: raise ValueError("n = 1 implies we have to test k = 0 only, and 0*b^1+/1 is either 1 or 1, which are not prime.") else: n = int(minn) while n <= int(maxn): #we'll set the max sieve depth via if/else statements, #we can adjust this if needed if n <= 5000: sievedepth = 1000000 elif n <= 10000: sievedepth = 5000000 elif n <= 20000: sievedepth = 25000000 elif n <= 40000: sievedepth = 100000000 else: sievedepth = 250000000 #calculate maxk, which is n1 maxk = n1 #for n = 2 we have to be a bit more careful. The only meaningful k is 1. But twinsieve gives a error: kmax has to be greater than kmin. #so we will tell the user that he'll need to test it himself with pfgw. if n == 2: print("n = 2 yields an error in twinsieve. You'll need to test " + str(b) + "^" + str(n) + "+/1 yourself in LLR or pfgw.") n = n + 1 else: #now call subprocess. subprocess.run(["twinsieve", "P", str(sievedepth), "k", "1", "K", str(maxk), "b", str(b),"n", str(n)]) n = n+1 Code:
3*20^8+1 3*20^81 105*20^152+1 105*20^1521 24*20^36+1 24*20^361 60*20^68+1 60*20^681 3*20^69+1 3*20^691 

20200611, 20:13  #257 
"Sam"
Nov 2016
311 Posts 
I don't suppose newpgen + pfgw would be faster than twinsieve ?
Here are the twin primes bases up to 48: Code:
k*b^n+1 with k <= n base = 3 (check to n=15000) 2*3^2+1 8*3^10+1 4*3^15+1 10*3^22+1 10*3^102+1 76*3^139+1 928*3^988+1 476*3^1483+1 2618*3^4286+1 2926*3^11071+1  base = 5 (check to n=2000) 12*5^51+1 84*5^103+1 156*5^202+1 336*5^765+1  base = 6 (check to n=2000) 1*6^1+1 2*6^2+1 2*6^3+1 2*6^4+1 3*6^6+1 17*6^35+1 23*6^67+1 143*6^162+1 187*6^251+1 152*6^279+1 157*6^371+1 257*6^824+1 430*6^1318+1 1743*6^1916+1 613*6^1922+1  base = 7 (check to n=2000) (none)  base = 10 (check to n=2000) 3*10^3+1 3*10^7+1 126*10^182+1 525*10^632+1  base = 11 (check to n=2000) 1182*11^1409+1  base = 12 (check to n=2000) 1*12^1+1 4*12^5+1 4*12^15+1 860*12^967+1  base = 13 (check to n=2000) 180*13^202+1 228*13^428+1  base = 14 (check to n=2000) (none)  base = 15 (check to n=2000) 2*15^10+1 14*15^14+1 2*15^20+1 238*15^353+1  base = 17 (check to n=2000) (none)  base = 18 (check to n=2000) 1*18^1+1 9*18^11+1 231*18^307+1 357*18^1664+1  base = 19 (check to n=2000) (none)  base = 20 (check to n=2000) 24*20^36+1 3*20^69+1 105*20^152+1  base = 21 (check to n=2000) 8*21^26+1 22*21^26+1 30*21^44+1 52*21^55+1 418*21^1919+1  base = 22 (check to n=2000) (none)  base = 23 (check to n=2000) (none)  base = 24 (check to n=2000) 13*24^23+1 10*24^66+1  base = 26 (check to n=2000) 210*26^742+1 837*26^1244+1  base = 28 (check to n=2000) 12*28^16+1  base = 29 (check to n=2000) (none)  base = 30 (check to n=2000) 1*30^1+1 14*30^43+1 141*30^169+1 14*30^262+1 446*30^504+1 1389*30^1563+1  base = 31 (check to n=2000) 168*31^183+1  base = 33 (check to n=2000) (none)  base = 34 (check to n=2000) 3*34^11+1 255*34^676+1 828*34^856+1  base = 35 (check to n=2000) 930*35^1167+1  base = 37 (check to n=2000) (none)  base = 38 (check to n=2000) 3*38^10+1 9*38^53+1 45*38^111+1  base = 39 (check to n=2000) 608*39^706+1  base = 40 (check to n=2000) 30*40^39+1 3*40^324+1 273*40^326+1 132*40^574+1  base = 41 (check to n=2000) 168*41^261+1 312*41^1208+1  base = 42 (check to n=2000) 1*42^1+1 5*42^9+1 6*42^57+1 90*42^121+1 53*42^158+1 652*42^746+1  base = 43 (check to n=2000) 30*43^1525+1  base = 44 (check to n=2000) 3*44^9+1  base = 45 (check to n=2000) 2*45^8+1 84*45^84+1 268*45^318+1 136*45^768+1 308*45^970+1  base = 46 (check to n=2000) 18*46^25+1 267*46^358+1  base = 47 (check to n=2000) (none)  base = 48 (check to n=2000) 4*48^7+1 2*48^8+1 3*48^8+1 24*48^323+1 30*48^673+1  Last fiddled with by carpetpool on 20200611 at 20:14 
20200611, 20:26  #258 
"Dylan"
Mar 2017
2·257 Posts 
I'd imagine for small n and b newpgen and twinsieve will take roughly the same time. For larger values of these quantities twinsieve will likely have the advantage as 1. It doesn't have the memory restrictions that newpgen has, and 2. It's part of the mtsieve framework, so we can run it multithreaded.
And it appears your list for b = 20 is missing two primes: the ones for n = 8 (k value is 3) and 68 (k value is 60). 
20200615, 21:51  #259 
"Sam"
Nov 2016
311 Posts 
Here is the complete set: Bases <= 24 checked to n=5K, others < 100 checked to n=2K. Also verified smaller twin primes, which I had forgot most of them in my previous list.

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