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
1427194 · 113427194 + 1 877069 p310 Jan 2012 Generalized Cullen
2400254 · 127400254 + 1 842062 g407 Jun 2013 Generalized Cullen
3374565 · 22247391 + 1 676538 L3532 Jun 2013 Generalized Cullen
4292402 · 159292402 + 1 643699 g407 Nov 2012 Generalized Cullen
5437960 · 31313880 + 1 626886 L2777 Nov 2012 Generalized Cullen
6269328 · 211269328 + 1 626000 p354 Jun 2012 Generalized Cullen
71286 · 3937499 + 1 447304 L2777 Feb 2012 Generalized Cullen
894189 · 21318646 + 1 396957 L2777 Feb 2013 Generalized Cullen
9259738 · 3779214 + 1 371785 L2777 Dec 2011 Generalized Cullen
10177482 · 117177482 + 1 367072 g407 Feb 2008 Generalized Cullen
11183500 · 93183500 + 1 361222 g157 Oct 2012 Generalized Cullen
12174885 · 98174885 + 1 348241 g157 Aug 2012 Generalized Cullen
13178192 · 3712768 + 1 340083 L2777 Dec 2011 Generalized Cullen
1414521 · 6435631 + 1 338991 L2777 Jan 2012 Generalized Cullen
15139948 · 151139948 + 1 304949 g407 Mar 2010 Generalized Cullen
16236418 · 17236418 + 1 290906 g157 Dec 2012 Generalized Cullen
17143717 · 96143717 + 1 284892 g157 Apr 2009 Generalized Cullen
18122095 · 198122095 + 1 280417 g407 Nov 2012 Generalized Cullen
19138846 · 99138846 + 1 277092 g157 Jun 2010 Generalized Cullen
2042014 · 3504169 + 1 240555 L2777 Nov 2011 Generalized Cullen
Chris K. Caldwell © 1996-2013 (all rights reserved)