The Top Twenty--a Prime Page Collection

Generalized Cullen

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The Prime Pages

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

(up) Definitions and Notes

In 1905, the Reverend Cullen was interested in the numbers n.2n+1 (denoted Cn).  He noticed that the first, C1=3, was prime, but with the possible exception of the 53rd, the next 99 were all composite.  Very soon afterwards, Cunningham discovered that 5591 divides C53, and noted these numbers are composite for all n in the range 2 < n < 200, with the possible exception of 141.  Five decades later Robinson showed C141 was a prime.

The Generalized Cullen primes are the primes of the form n.bn+1 with n+2 > b. The reason for the restriction on the exponent n is simple, without some restriction every prime p would be a generalized Cullen because:

p = 1.(p-1)1+1.
Curiously, these numbers may be hard to recognize when written in standard form.  For example, they may be like
72048*10144096+1
which could be written
72048*10072048+1.
More difficult to spot are those like the following:
39284*3235705+1 = (39284*3)*3235704+1 = 117852*9117852+1
669*2128454+1 = (669*26)*2128448+1 = 42816*842816+1.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1177482 · 117177482+1 367072 g407 Feb 2008 Generalized Cullen
2143717 · 96143717+1 284892 g157 Apr 2009 Generalized Cullen
3101670 · 91101670+1 199181 g157 May 2005 Generalized Cullen
487258 · 18287258+1 197215 g392 Mar 2006 Generalized Cullen
598035 · 10196070+1 196075 g157 Feb 2007 Generalized Cullen
637510 · 3337592+1 161077 p126 Jun 2006 Generalized Cullen
792278 · 5092278+1 156783 g157 Dec 2008 Generalized Cullen
856271 · 2506439+1 152459 p243 Jan 2009 Generalized Cullen
981778 · 6681778+1 148804 g157 Nov 2007 Generalized Cullen
1072048 · 10144096+1 144101 g157 May 2005 Generalized Cullen
1188195 · 4088195+1 141299 x37 May 2009 Generalized Cullen
1264872 · 14564872+1 140218 g142 Apr 2005 Generalized Cullen
1391848 · 3391848+1 139478 g157 Dec 2006 Generalized Cullen
1461813 · 17261813+1 138190 g407 Aug 2007 Generalized Cullen
1576710 · 6176710+1 136958 g157 Jul 2006 Generalized Cullen
1662378 · 14162378+1 134069 g407 Jun 2007 Generalized Cullen
1774460 · 5974460+1 131863 g157 Jul 2006 Generalized Cullen
1858897 · 16658897+1 130763 g407 Mar 2007 Generalized Cullen
1988900 · 2688900+1 125797 g157 Jun 2005 Generalized Cullen
2070615 · 6070615+1 125569 g157 Feb 2008 Generalized Cullen
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