Cullen-Williams primes and Woodall-Williams primes
 

The [B]Cullen-Williams number[/B] 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 [B]Woodall-Williams number[/B] 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 [B]Cullen-Williams number[/B] 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 [B]Woodall-Williams number[/B] 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?

Do you have search limits for these forms?

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 <whatever limit you want>

Running to a higher value to see if anything else shows up.

Faster still is to use plain ABC.
[C]ABC2 ($a-1)*$a^($a-1)+1[/C] uses generic FFT.

Instead, run something like this:
[C]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[/C]

OEIS: [OEIS]271718[/OEIS] , [OEIS]191568[/OEIS]

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.

Continuing Woodall-Williams from b=12000.