## 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. Comments and suggestions requested.

In 1905, the Reverend Cullen was interested in the numbers
*n*^{.}2^{n}+1 (denoted C_{n}).
He noticed that the first, C_{1}=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 C_{53}, 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 C_{141} was a prime.

The **Generalized Cullen primes** are the primes of
the form *n*^{.}*b*^{n}+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:

Curiously, these numbers may be hard to recognize when written in standard form. For example, they may be likep= 1^{.}(p-1)^{1}+1.

72048*10which could be written^{144096}+1

72048*100More difficult to spot are those like the following:^{72048}+1.

39284*3^{235705}+1 = (39284*3)*3^{235704}+1 = 117852*9^{117852}+1

669*2^{128454}+1 = (669*2^{6})*2^{128448}+1 = 42816*8^{42816}+1.

rank prime digits who when comment 1 2805222 · 5^{5610444}+ 13921539 L4972 Sep 2019 Generalized Cullen 2 1806676 · 41^{1806676}+ 12913785 L4668 Mar 2018 Generalized Cullen 3 1323365 · 116^{1323365}+ 12732038 L4718 Jan 2018 Generalized Cullen 4 1341174 · 53^{1341174}+ 12312561 L4668 Aug 2017 Generalized Cullen 5 682156 · 79^{682156}+ 11294484 L4472 Oct 2016 Generalized Cullen 6 298989 · 2^{3886857}+ 11170067 L2777 Dec 2014 Generalized Cullen 7 27777 · 2^{3111027}+ 1936517 L2777 Feb 2014 Generalized Cullen 8 46425 · 2^{2971203}+ 1894426 L2777 Feb 2014 Generalized Cullen 9 427194 · 113^{427194}+ 1877069 p310 Jan 2012 Generalized Cullen 10 400254 · 127^{400254}+ 1842062 g407 Jun 2013 Generalized Cullen 11 374565 · 2^{2247391}+ 1676538 L3532 Jun 2013 Generalized Cullen 12 292402 · 159^{292402}+ 1643699 g407 Nov 2012 Generalized Cullen 13 316903 · 10^{633806}+ 1633812 L3532 Jul 2014 Generalized Cullen 14 437960 · 3^{1313880}+ 1626886 L2777 Nov 2012 Generalized Cullen 15 269328 · 211^{269328}+ 1626000 p354 Jun 2012 Generalized Cullen 16 1183414 · 3^{1183414}+ 1564639 L2841 Jan 2014 Generalized Cullen 17 1286 · 3^{937499}+ 1447304 L2777 Feb 2012 Generalized Cullen 18 94189 · 2^{1318646}+ 1396957 L2777 Feb 2013 Generalized Cullen 19 259738 · 3^{779214}+ 1371785 L2777 Dec 2011 Generalized Cullen 20 177482 · 117^{177482}+ 1367072 g407 Feb 2008 Generalized Cullen

Chris K. Caldwell
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