20201024, 16:28  #1 
Nov 2016
2^{2}·691 Posts 
CullenWilliams primes and WoodallWilliams primes
The CullenWilliams number base b is (b1)*b^(b1)+1, which is both Cullen number base b (n*b^n+1, some author requires n>=b1, and for this number n is exactly b1) and 2nd Williams number base b ((b1)*b^n+1)
The WoodallWilliams number base b is (b1)*b^(b1)1, which is both Woodall number base b (n*b^n1, some author requires n>=b1, and for this number n is exactly b1) and 1st Williams number base b ((b1)*b^n1) The CullenWilliams number base b, (b1)*b^(b1)+1 is prime for b = 2, 3, 4, 10, 11, 15, 34, 37, ... (they are exactly the smallest Cullen prime base b for b = 2, 3, 11, 37, and they are exactly the smallest 2nd Williams prime base b for b = 2 and 11) The WoodallWilliams number base b, (b1)*b^(b1)1 is prime for 3, 4, 8, 15, 44, 82, ... (they are exactly the smallest Woodall prime base b for b = 82, and they are exactly the smallest 2nd Williams prime base b for b = 15 and 82) What are the next CullenWilliams prime and the next WoodallWilliams prime? 
20201024, 19:24  #2 
"Dylan"
Mar 2017
547 Posts 
Do you have search limits for these forms?

20201025, 20:23  #3 
"Mark"
Apr 2003
Between here and the
1100000011001_{2} Posts 
Must not be too deeply searched. A pfgw script to b = 1000 yields the PRPs (9441)*944^(9441)1 and (16221)*1622^(16221)1
Here is the script. Use f to trial factor before PRP testing. ABC2 ($a1)*$a^($a1)+1  ($a1)*$a^($a1)1 a: from 1 to <whatever limit you want> Running to a higher value to see if anything else shows up. Last fiddled with by rogue on 20201025 at 20:24 
20201025, 23:35  #4 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
47×197 Posts 

20201026, 12:03  #5 
"Mark"
Apr 2003
Between here and the
31×199 Posts 
I stopped searching at b=12000 and am stopping. Someone else can take it further.
There *might* be value in someone using sr1sieve with a script to find factors rather than using pfgw to find factors. Last fiddled with by rogue on 20201026 at 12:04 
20201028, 00:36  #6 
"W. Byerly"
Aug 2013
1423*2^21790231
3^{2}·11 Posts 
Continuing WoodallWilliams from b=12000.

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