## Generalized Cullen |

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 404849 · 2^{13764867}+ 14143644 L4976 Mar 2021 Generalized Cullen 2 2805222 · 5^{5610444}+ 13921539 L4972 Sep 2019 Generalized Cullen 3 1806676 · 41^{1806676}+ 12913785 L4668 Mar 2018 Generalized Cullen 4 1323365 · 116^{1323365}+ 12732038 L4718 Jan 2018 Generalized Cullen 5 1341174 · 53^{1341174}+ 12312561 L4668 Aug 2017 Generalized Cullen 6 682156 · 79^{682156}+ 11294484 L4472 Oct 2016 Generalized Cullen 7 298989 · 2^{3886857}+ 11170067 L2777 Dec 2014 Generalized Cullen 8 27777 · 2^{3111027}+ 1936517 L2777 Feb 2014 Generalized Cullen 9 46425 · 2^{2971203}+ 1894426 L2777 Feb 2014 Generalized Cullen 10 427194 · 113^{427194}+ 1877069 p310 Jan 2012 Generalized Cullen 11 400254 · 127^{400254}+ 1842062 g407 Jun 2013 Generalized Cullen 12 374565 · 2^{2247391}+ 1676538 L3532 Jun 2013 Generalized Cullen 13 292402 · 159^{292402}+ 1643699 g407 Nov 2012 Generalized Cullen 14 316903 · 10^{633806}+ 1633812 L3532 Jul 2014 Generalized Cullen 15 437960 · 3^{1313880}+ 1626886 L2777 Nov 2012 Generalized Cullen 16 269328 · 211^{269328}+ 1626000 p354 Jun 2012 Generalized Cullen 17 1183414 · 3^{1183414}+ 1564639 L2841 Jan 2014 Generalized Cullen 18 1286 · 3^{937499}+ 1447304 L2777 Feb 2012 Generalized Cullen 19 94189 · 2^{1318646}+ 1396957 L2777 Feb 2013 Generalized Cullen 20 259738 · 3^{779214}+ 1371785 L2777 Dec 2011 Generalized Cullen

Chris K. Caldwell
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