mersenneforum.org Cullen-Williams primes and Woodall-Williams primes
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 2020-10-24, 16:28 #1 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 C7C16 Posts Cullen-Williams primes and Woodall-Williams primes The Cullen-Williams number base b is (b-1)*b^(b-1)+1, which is both Cullen number base b (n*b^n+1, some author requires n>=b-1, and for this number n is exactly b-1) and 2nd Williams number base b ((b-1)*b^n+1) The Woodall-Williams number base b is (b-1)*b^(b-1)-1, which is both Woodall number base b (n*b^n-1, some author requires n>=b-1, and for this number n is exactly b-1) and 1st Williams number base b ((b-1)*b^n-1) The Cullen-Williams number base b, (b-1)*b^(b-1)+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 Woodall-Williams number base b, (b-1)*b^(b-1)-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 Cullen-Williams prime and the next Woodall-Williams prime?
 2020-10-24, 19:24 #2 Dylan14     "Dylan" Mar 2017 22×3×72 Posts Do you have search limits for these forms?
 2020-10-25, 20:23 #3 rogue     "Mark" Apr 2003 Between here and the 22×7×233 Posts Must not be too deeply searched. A pfgw script to b = 1000 yields the PRPs (944-1)*944^(944-1)-1 and (1622-1)*1622^(1622-1)-1 Here is the script. Use -f to trial factor before PRP testing. ABC2 ($a-1)*$a^($a-1)+1 | ($a-1)*$a^($a-1)-1 a: from 1 to Running to a higher value to see if anything else shows up. Last fiddled with by rogue on 2020-10-25 at 20:24
 2020-10-25, 23:35 #4 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 22×32×269 Posts Faster still is to use plain ABC. ABC2 ($a-1)*$a^($a-1)+1 uses generic FFT. Instead, run something like this: cat > a1.abc ABC$a*$b^$c$d ^D seq 1 20000 | awk '{print$1-1,$1,$1-1,"+1"}' >> a1.abc pfgw64 -N -f -l a1.abc OEIS: 271718 , 191568
 2020-10-26, 12:03 #5 rogue     "Mark" Apr 2003 Between here and the 22·7·233 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 2020-10-26 at 12:04
 2020-10-28, 00:36 #6 Trilo     "W. Byerly" Aug 2013 81*2^2797443-1 5×23 Posts Continuing Woodall-Williams from b=12000.
 2021-12-01, 14:19 #7 Trilo     "W. Byerly" Aug 2013 81*2^2797443-1 7316 Posts Woodall-Williams Series is now to b=100000, no new primes. I wrote a (slow) python sieve for "Generalized Woodall-Williams/Cullen-Williams" 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 (b-1)*b^(b+1) +/- 1 both to 20000: (b-1)*b^(b+1) - 1 is prime for b= 1, 5, 18, 6073 (b-1)*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?
 2021-12-01, 17:39 #8 kar_bon     Mar 2006 Germany 1011011100012 Posts 17*18^19-1 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: (b-1)*b^(b+1) - 1 is prime for b= 1, 2, 6, 19, 6073. and (b-1)*b^(b+1) + 1 is prime for b= 3, 5. Last fiddled with by kar_bon on 2021-12-01 at 17:47 Reason: others
2021-12-02, 18:27   #9
rogue

"Mark"
Apr 2003
Between here and the

22·7·233 Posts

Quote:
 Originally Posted by Trilo Woodall-Williams Series is now to b=100000, no new primes. I wrote a (slow) python sieve for "Generalized Woodall-Williams/Cullen-Williams" 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 (b-1)*b^(b+1) +/- 1 both to 20000: (b-1)*b^(b+1) - 1 is prime for b= 1, 5, 18, 6073 (b-1)*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?
Are you continuing to work on this? Can you share your script?

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