20201024, 16:28  #1 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3,389 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
250_{16} Posts 
Do you have search limits for these forms?

20201025, 20:23  #3 
"Mark"
Apr 2003
Between here and the
7×23×41 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
3×29×113 Posts 

20201026, 12:03  #5 
"Mark"
Apr 2003
Between here and the
7×23×41 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
81*2^31743531
1111110_{2} Posts 
Continuing WoodallWilliams from b=12000.

20211201, 14:19  #7 
"W. Byerly"
Aug 2013
81*2^31743531
2·3^{2}·7 Posts 
WoodallWilliams Series is now to b=100000, no new primes.
I wrote a (slow) python sieve for "Generalized WoodallWilliams/CullenWilliams" numbers of the form (b+x)*b^(b+y) +/ 1 for constant x, y (woodall williams series has x, y = 1.) using fbncsieve. If there is any interest I'll release the source code. Seached (b1)*b^(b+1) +/ 1 both to 20000: (b1)*b^(b+1)  1 is prime for b= 1, 5, 18, 6073 (b1)*b^(b+1) + 1 is prime for b= 2, 4. Who will be the first to find a number of this form large enough for the top 5000 list? 
20211201, 17:39  #8 
Mar 2006
Germany
2952_{10} Posts 
17*18^191 is not prime, so your b is 19 not 18. Same for b=5 and b=2 is also a prime for the first form.
So: (b1)*b^(b+1)  1 is prime for b= 1, 2, 6, 19, 6073. and (b1)*b^(b+1) + 1 is prime for b= 3, 5. Last fiddled with by kar_bon on 20211201 at 17:47 Reason: others 
20211202, 18:27  #9  
"Mark"
Apr 2003
Between here and the
7·23·41 Posts 
Quote:


20220407, 19:58  #10 
"W. Byerly"
Aug 2013
81*2^31743531
2×3^{2}×7 Posts 
b*(b+1)^b 1 is searched until 118000, no new primes.
let me clean up the code for my sieve a bit and I'll release it. Last fiddled with by Trilo on 20220407 at 19:58 
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