# 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
29145334 · 39145334 + 1 4363441 A6 Dec 2023 Generalized Cullen
3404849 · 213764867 + 1 4143644 L4976 Mar 2021 Generalized Cullen
45778486 · 55778486 + 1 4038996 A6 Aug 2024 Generalized Cullen
52805222 · 55610444 + 1 3921539 L4972 Sep 2019 Generalized Cullen
6763795 · 64582771 + 1 3566095 A6 Dec 2023 Generalized Cullen
74450871 · 64450871 + 1 3463458 L5765 Sep 2023 Generalized Cullen
8732050 · 64392301 + 1 3417881 L5765 Sep 2023 Generalized Cullen
91806676 · 411806676 + 1 2913785 L4668 Mar 2018 Generalized Cullen
101323365 · 1161323365 + 1 2732038 L4718 Jan 2018 Generalized Cullen
111341174 · 531341174 + 1 2312561 L4668 Aug 2017 Generalized Cullen
121909683 · 141909683 + 1 2188748 L5765 May 2023 Generalized Cullen
132240501 · 62240501 + 1 1743456 L5765 Jul 2023 Generalized Cullen
141486287 · 141486287 + 1 1703482 L5765 May 2023 Generalized Cullen
15884411 · 38884411 + 1 1397184 L5765 May 2023 Generalized Cullen
161198433 · 141198433 + 1 1373564 L5765 May 2023 Generalized Cullen
17682156 · 79682156 + 1 1294484 L4472 Oct 2016 Generalized Cullen
18993149 · 20993149 + 1 1292123 L5765 May 2023 Generalized Cullen
191644731 · 61644731 + 1 1279856 L5765 Jul 2023 Generalized Cullen
20298989 · 23886857 + 1 1170067 L2777 Dec 2014 Generalized Cullen