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Generalized Cullen/Woodall prime search :
Welcome (back) to the Generalized Cullen/Woodall Prime Search
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Michael GoetzVolunteer moderator Project administrator Project scientist
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Joined: 21 Jan 10 Posts: 13174 ID: 53948 Credit: 213,215,523 RAC: 186,459

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 is moving its search for Generalized Cullen and Generalized Woodall primes from PRPNet to BOINC. As is customary when projects move from PRPNet, PrimeGrid will doublecheck the ranges searched by PRPNet, and will then continue on with new work running multiple bases (b values) concurrently and incrementing through n values.
PrimeGrid will be sieving to a much larger n than has been previously done. The largest candidates will be in excess of 15,000,000 digits, and will be the same size as the largest candidates in the Seventeen or Bust project.
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.
Once the sieving has built a sufficient and sustainable pool of credits, PrimeGrid anticipates restarting LLR work as well, and would expect this to occur in early 2017.
In addition to having found the largest known Cullen prime http://primes.utm.edu/primes/page.php?id=89536 and largest known Woodall prime http://primes.utm.edu/primes/page.php?id=83407, PrimeGrid has found the largest known Generalized Cullen prime, http://primes.utm.edu/primes/page.php?id=124515 and the 4th largest known Generalized Woodall prime http://primes.utm.edu/primes/page.php?id=98862.
For more information on Generalized Cullen and Woodall Numbers, you can go here: http://primes.utm.edu/top20/page.php?id=42 and here: http://primes.utm.edu/top20/page.php?id=45.
 


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  


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?
____________
 


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*175^61.
/JeppeSN
 

rogueVolunteer developer
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Joined: 8 Sep 07 Posts: 1191 ID: 12001 Credit: 18,565,548 RAC: 0

(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.  


(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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Generalized Cullen/Woodall prime search :
Welcome (back) to the Generalized Cullen/Woodall Prime Search 