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Generalized Cullen
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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 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
| rank | prime |
digits | who | when | comment |
|---|
| 1 | 2525532 · 732525532 + 1 |
4705888 |
L5402 |
Aug 2021 |
Generalized Cullen |
| 2 | 404849 · 213764867 + 1 |
4143644 |
L4976 |
Mar 2021 |
Generalized Cullen |
| 3 | 2805222 · 55610444 + 1 |
3921539 |
L4972 |
Sep 2019 |
Generalized Cullen |
| 4 | 1806676 · 411806676 + 1 |
2913785 |
L4668 |
Mar 2018 |
Generalized Cullen |
| 5 | 1323365 · 1161323365 + 1 |
2732038 |
L4718 |
Jan 2018 |
Generalized Cullen |
| 6 | 1341174 · 531341174 + 1 |
2312561 |
L4668 |
Aug 2017 |
Generalized Cullen |
| 7 | 682156 · 79682156 + 1 |
1294484 |
L4472 |
Oct 2016 |
Generalized Cullen |
| 8 | 298989 · 23886857 + 1 |
1170067 |
L2777 |
Dec 2014 |
Generalized Cullen |
| 9 | 27777 · 23111027 + 1 |
936517 |
L2777 |
Feb 2014 |
Generalized Cullen |
| 10 | 46425 · 22971203 + 1 |
894426 |
L2777 |
Feb 2014 |
Generalized Cullen |
| 11 | 427194 · 113427194 + 1 |
877069 |
p310 |
Jan 2012 |
Generalized Cullen |
| 12 | 400254 · 127400254 + 1 |
842062 |
g407 |
Jun 2013 |
Generalized Cullen |
| 13 | 374565 · 22247391 + 1 |
676538 |
L3532 |
Jun 2013 |
Generalized Cullen |
| 14 | 292402 · 159292402 + 1 |
643699 |
g407 |
Nov 2012 |
Generalized Cullen |
| 15 | 316903 · 10633806 + 1 |
633812 |
L3532 |
Jul 2014 |
Generalized Cullen |
| 16 | 437960 · 31313880 + 1 |
626886 |
L2777 |
Nov 2012 |
Generalized Cullen |
| 17 | 269328 · 211269328 + 1 |
626000 |
p354 |
Jun 2012 |
Generalized Cullen |
| 18 | 1183414 · 31183414 + 1 |
564639 |
L2841 |
Jan 2014 |
Generalized Cullen |
| 19 | 1286 · 3937499 + 1 |
447304 |
L2777 |
Feb 2012 |
Generalized Cullen |
| 20 | 94189 · 21318646 + 1 |
396957 |
L2777 |
Feb 2013 |
Generalized Cullen |
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