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.

This page is about one of those forms.

Definitions and Notes

In 1905, the Reverend Cullen was interested in the numbers n^.2ⁿ+1 (denoted C_n).  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

rank prime digits who when comment 1 4052186 · 694052186 + 1 7451366 A61 Apr 2025 Generalized Cullen 2 2525532 · 732525532 + 1 4705888 L5402 Aug 2021 Generalized Cullen 3 9145334 · 39145334 + 1 4363441 A6 Dec 2023 Generalized Cullen 4 404849 · 213764867 + 1 4143644 L4976 Mar 2021 Generalized Cullen 5 5778486 · 55778486 + 1 4038996 A6 Aug 2024 Generalized Cullen 6 2805222 · 55610444 + 1 3921539 L4972 Sep 2019 Generalized Cullen 7 763795 · 64582771 + 1 3566095 A6 Dec 2023 Generalized Cullen 8 4450871 · 64450871 + 1 3463458 L5765 Sep 2023 Generalized Cullen 9 732050 · 64392301 + 1 3417881 L5765 Sep 2023 Generalized Cullen 10 1806676 · 411806676 + 1 2913785 L4668 Mar 2018 Generalized Cullen 11 1323365 · 1161323365 + 1 2732038 L4718 Jan 2018 Generalized Cullen 12 1341174 · 531341174 + 1 2312561 L4668 Aug 2017 Generalized Cullen 13 1909683 · 141909683 + 1 2188748 L5765 May 2023 Generalized Cullen 14 2240501 · 62240501 + 1 1743456 L5765 Jul 2023 Generalized Cullen 15 1486287 · 141486287 + 1 1703482 L5765 May 2023 Generalized Cullen 16 884411 · 38884411 + 1 1397184 L5765 May 2023 Generalized Cullen 17 1198433 · 141198433 + 1 1373564 L5765 May 2023 Generalized Cullen 18 682156 · 79682156 + 1 1294484 L4472 Oct 2016 Generalized Cullen 19 993149 · 20993149 + 1 1292123 L5765 May 2023 Generalized Cullen 20 1644731 · 61644731 + 1 1279856 L5765 Jul 2023 Generalized Cullen