# Generalized Cullen

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.

### 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.

### Record Primes of this Type

rankprime digitswhowhencomment
12525532 · 732525532 + 1 4705888 L5402 Aug 2021 Generalized Cullen
2404849 · 213764867 + 1 4143644 L4976 Mar 2021 Generalized Cullen
32805222 · 55610444 + 1 3921539 L4972 Sep 2019 Generalized Cullen
41806676 · 411806676 + 1 2913785 L4668 Mar 2018 Generalized Cullen
51323365 · 1161323365 + 1 2732038 L4718 Jan 2018 Generalized Cullen
61341174 · 531341174 + 1 2312561 L4668 Aug 2017 Generalized Cullen
7682156 · 79682156 + 1 1294484 L4472 Oct 2016 Generalized Cullen
8298989 · 23886857 + 1 1170067 L2777 Dec 2014 Generalized Cullen
927777 · 23111027 + 1 936517 L2777 Feb 2014 Generalized Cullen
1046425 · 22971203 + 1 894426 L2777 Feb 2014 Generalized Cullen
11427194 · 113427194 + 1 877069 p310 Jan 2012 Generalized Cullen
12400254 · 127400254 + 1 842062 g407 Jun 2013 Generalized Cullen
13374565 · 22247391 + 1 676538 L3532 Jun 2013 Generalized Cullen
14292402 · 159292402 + 1 643699 g407 Nov 2012 Generalized Cullen
15316903 · 10633806 + 1 633812 L3532 Jul 2014 Generalized Cullen
16437960 · 31313880 + 1 626886 L2777 Nov 2012 Generalized Cullen
17269328 · 211269328 + 1 626000 p354 Jun 2012 Generalized Cullen
181183414 · 31183414 + 1 564639 L2841 Jan 2014 Generalized Cullen
191286 · 3937499 + 1 447304 L2777 Feb 2012 Generalized Cullen
2094189 · 21318646 + 1 396957 L2777 Feb 2013 Generalized Cullen