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 moved its search for Generalized Cullen and Generalized Woodall primes from PRPNet to BOINC in September 2016. As is customary when projects move from PRPNet, PrimeGrid double checked the ranges searched by PRPNet, and is currently continuing with new work running multiple bases (b values) concurrently, with incrementing n values.

PrimeGrid will be sieving to a much larger n than has beenpreviously 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, at n=50M.

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 smaller than 130, PrimeGrid has found a Generalized Woodall prime, and will now be searching only for Generalized Cullen Primes of certain bases.

The following bases have yet to produce a prime (highlighted bases have been eliminated):

  • Woodall b=43104 & 121
  • Cullen b=13, 25, 29, 41, 47, 49, 53, 55, 68, 69, 73, 79, 101, 109, 113116 & 121

Base 149 is the next primeless base for both generalized Cullen and Generalized Woodall.

The GCW Sieve continued until May 1st, 2019, when it reached optimal on all bases. LLR work on GCW restarted in 2017.

In addition to having found the largest known Cullen prime at http://primes.utm.edu/primes/page.php?id=89536 and the largest known Woodall prime at http://primes.utm.edu/primes/page.php?id=124539,

PrimeGrid has found the largest known Generalized Cullen prime, http://primes.utm.edu/primes/page.php?id=129893 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.

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