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Generalized Cullen/Woodall prime search :
Welcome to the Generalized Cullen/Woodall Prime Search
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 Michael Goetz  Volunteer moderator
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A Cullen number (first studied by Reverend James Cullen in 1905) is a number of the form n * 2^n + 1. A Woodall number (first studied by Allan Cunningham and H.J. Woodall in 1917) is a number of the form n * 2^n - 1.
Generalized Cullen and Woodall numbers are of the form n * b^n + 1 and n * b^n - 1, respectively, where n + 2 > b.
PrimeGrid has moved its search for Generalized Cullen and Generalized Woodall primes from PRPNet to BOINC. As is customary when projects move from PRPNet, PrimeGrid has double-checked the ranges searched by PRPNet, and then continued on with new work running multiple bases (b values) concurrently and incrementing through n values.
PrimeGrid has completed sieving to a much larger n than was previously done. The largest candidates are in excess of 15,000,000 digits, and will be the same size as the largest candidates in the Seventeen or Bust project as of the time of this writing.
Once PrimeGrid finds a Generalized Cullen or Woodall on a base, it stops looking for Generalized Cullen or Woodall primes on that base, depending on the type found. For all the current bases, PrimeGrid has found a Generalized Woodall prime, and will initially be searching only for Generalized Cullen Primes.
The following bases have yet to produce a prime (highlighted ones have been found):
- Woodall b=43, 104 & 121
- Cullen b=13, 25, 29, 41, 47, 49, 53, 55, 68, 69, 73, 79, 101, 109, 113, 116 & 121
Base 149 is the next primeless base for both GC and GW.
In addition to having found the largest known Cullen prime https://t5k.org/primes/page.php?id=89536 and largest known Woodall prime https://t5k.org/primes/page.php?id=83407, PrimeGrid has found the largest known Generalized Cullen prime, https://t5k.org/primes/page.php?id=140607 and the 4th largest known Generalized Woodall prime https://t5k.org/primes/page.php?id=98862.
For more information on Generalized Cullen and Woodall Numbers, you can go here: https://t5k.org/top20/page.php?id=42 and here: https://t5k.org/top20/page.php?id=45.
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I found some lists with known n values for each b:
* Günter Löh (generalized Cullens with 3≤b≤100)
* Steven Harvey (generalized Woodalls with 3≤b≤10000, and generalized Cullens with 101≤b≤10000, and more)
Be aware of the requirement n > b - 2. From Löh's list, it looks like, for generalized Cullens, the bases b=11 and b=37 are not "resolved" if we strengthen the requirement to n > b.
/JeppeSN | |
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Be aware of the requirement n > b - 2. From Löh's list, it looks like, for generalized Cullens, the bases b=11 and b=37 are not "resolved" if we strengthen the requirement to n > b.
What would be the reasoning behind strengthening this requirement?
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Looking at these lists again, we have that:
- Cullen b=32, 75, 106, 115, ...
are bases for which no generalized Cullen is known if we really require n + 2 > b (as in the first post by Michael Goetz above).
It is easy to find:
- 5*32^5+1, 2*75^2+1, 3*106^3+1, 24*115^24+1, ...
but these do not meet the requirement n + 2 > b.
So maybe we should search these bases? What do you think?
(Question: Why does Löh say b=32 and b=64 are reserved by PrimeGrid?)
For a similar example with g. Woodall, b=175 only has 6*175^6-1.
/JeppeSN
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rogueVolunteer developer
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(Question: Why does Löh say b=32 and b=64 are reserved by PrimeGrid?)
At first I thought it was covered by the GFN search, but that couldn't be the case. | |
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(Question: Why does Löh say b=32 and b=64 are reserved by PrimeGrid?)
Because bases that are powers of 2 are covered by the GFN prime searches.
No, for example 100001*32^100001 + 1 = 100001*2^500005 + 1 is not a GFN. /JeppeSN | |
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It is interesting, are you going to eliminate 73 from your first comment in this thread? | |
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Generalized Cullen/Woodall prime search :
Welcome to the Generalized Cullen/Woodall Prime Search |
