[QUOTE=Dylan14;521757]I have searched n = 49796-49999 through to k = 1M (using twinsieve from the mtsieve suite and LLR) and found no twins. Hence the k/n pair for 49000-49999 is the right one.
If you need residues from that search, I have them.[/QUOTE]
Thanks! All we need now to complete the table is a smallest first twin k for n=44000-44999 and n=46000-46999. I'll plot a regression curve on it once we have that data.

[code]
Range Smallest First Twin k n-value
1000-1999 177 1032
2000-2999 4359 2191
3000-3999 1149 3283
4000-4999 2565 4901
5000-5999 5775 5907
6000-6999 4737 6634
7000-7999 33957 7768
8000-8999 459 8529
9000-9999 33891 9869
10000-10999 10941 10601
11000-11999 915 11455
12000-12999 73005 12178
13000-13999 3981 13153
14000-14999 175161 14171
15000-15999 74193 15770
16000-16999 138153 16436
17000-17999 14439 17527
18000-18999 56361 18989
19000-19999 53889 19817
20000-20999 7485 20023
21000-21999 195045 21432
22000-22999 31257 22312
23000-23999 396213 23672
24000-24999 177141 24365
25000-25999 577065 25879
26000-26999 182697 26172
27000-27999 70497 27652
28000-28999 445569 28353
29000-29999 815751 29705
30000-30999 249435 30977
31000-31999 440685 31989
32000-32999 51315 32430
33000-33999 143835 33826
34000-34999 959715 34895
35000-35999 338205 35351
36000-36999 47553 36172
37000-37999 201843 37630
38000-38999 683145 38746
39000-39999 126423 39606
40000-40999 604329 40315
41000-41999 358965 41653
42000-42999 272139 42379
43000-43999 441201 43167
44000-44999 >1M ???
45000-45999 311541 45439
46000-46999 >1M ???
47000-47999 103893 47122
48000-48999 694599 48501
49000-49999 197109 49733
[/code]

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[/CODE]

I didn't find any for base 7, although I'm sure they exist. The idea is that if k < n, we can get k as small as possible, so if all bases < 100 were tested, odds are you'll find a twin with a very small k. Continuing on with the search.

[QUOTE=carpetpool;547567]Also another interesting problem if anyone's interested:

Twin primes of the form k*b^n+-1 with k < n -->
[/QUOTE]

I wrote a Python script to drive the sieving:
[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 n-1
maxk = n-1
#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]Using this, I tested b = 20 to n = 2k. I found the following twin primes:
[CODE]3*20^8+1
3*20^8-1
105*20^152+1
105*20^152-1
24*20^36+1
24*20^36-1
60*20^68+1
60*20^68-1
3*20^69+1
3*20^69-1[/CODE]

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
---
[/CODE]

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).

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.

Reviving an old effort:

Here is an updated web page for all k*2^n-+1 twins for k<100K and n<=50K:
[URL]http://www.noprimeleftbehind.net/gary/twins100K.htm[/URL]

Here is an updated web page for all k*2^n-+1 twins for k=100K-1M and n<=50K if the k has a twin for n>10K:
[URL]http://www.noprimeleftbehind.net/gary/twins1M.htm[/URL]

Many years ago I had completed this effort to n=48K. I recently completed n=48K-50K. Here are the twins found in that range:
694599*2^48501-+1
852861*2^49195-+1
197109*2^49733-+1

I saw in this thread that at least 2 of those twins had already been found but this effort fully completes that search range.

k<1M is now fully complete to n=50K. The search depth of many other k's was updated to reflect mostly current efforts of various projects. No other twins were found. Mainly these included k<2000.

I have fully sieved files up to n=60K. My plan is to work on it on and off over the next few months.

Gary

k = 110.001 - 119.999, n = 110.001 - 120.000, twins = 0
 

Hello.
I`ve finished the range [B]k = 110.001 - 119.999 & n = 110.001 - 120.000[/B] for [url]http://prothsearch.com[/url] last year (xGF).
1181 Proth primes were found.
Now I`ve checked them to k*2^n-1 : [U]no twins[/U] (the file is attached)

I hope, it will be usefull stat.

P.S. I'll finish another range [B]k = 120.001 - 129.999 & n = 100.001 - 120.000[/B] in several months.

k = 115059, 118305, 126423 upto n = 200K
 

[QUOTE=gd_barnes;578449]
Here is an updated web page for all k*2^n-+1 twins for k=100K-1M and n<=50K if the k has a twin for n>10K:
[URL]http://www.noprimeleftbehind.net/gary/twins1M.htm[/URL][/QUOTE]

Gary,
I`ve doublechecked and tested [B]k = 115059, 118305, 126423[/B] (they are near my ranges) upto [B]n = 200K[/B].
[U]No new twins[/U].

I`ve found only 1 Proth prime (~4500 Proth-Riesel pairs of candidates were tested):
[QUOTE]118305*2^169842+1 is 3-PRP! (92.6915s+0.0008s)
118305*2^169842-1 is composite: RES64: [350692DEE5B9A708] (129.1531s+0.0373s)[/QUOTE]
[U]No new xGFs[/U].