mersenneforum.org Cullen-Williams primes and Woodall-Williams primes
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 2020-10-24, 16:28 #1 sweety439   Nov 2016 22·3·5·47 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 2×281 Posts Do you have search limits for these forms?
 2020-10-25, 20:23 #3 rogue     "Mark" Apr 2003 Between here and the 11000011001012 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 24×11×53 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 5×1,249 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 1423*2^2179023-1 103 Posts Continuing Woodall-Williams from b=12000.

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