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Old 2020-10-24, 16:28   #1
sweety439
 
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Default 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?
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Old 2020-10-24, 19:24   #2
Dylan14
 
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Do you have search limits for these forms?
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Old 2020-10-25, 20:23   #3
rogue
 
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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.

Last fiddled with by rogue on 2020-10-25 at 20:24
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Old 2020-10-25, 23:35   #4
Batalov
 
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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
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Old 2020-10-26, 12:03   #5
rogue
 
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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
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Old 2020-10-28, 00:36   #6
Trilo
 
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Continuing Woodall-Williams from b=12000.
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