# Generalized Repunit

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

A repunit is a number whose expansion (in radix 10) is a string of ones (for example: 11 and 11111111). A generalized repunit (radix b) is one whose expansion base b is all ones. For example, the Mersenne primes are the generalized repunits in base two (binary). Here is a formula for the n "digit" generalized repunit (base b):
(bn - 1)/(b - 1).
Clearly a generalized repunit prime is a generalized repunit that is prime.

In the special case that b is prime, the generalized repunit is the value of the sum of divisors function: σ (bn-1).

Note that every odd prime p is a generalized repunit in base p-1, in fact in that base it is "11". So we do not archive all generalized repunit primes, just those for which the number of digits is at least one fifth of the base (and the base must be positive!) This anthropocentric limit was choosen so that 11 (in base 10) is still considered a generalized repunit (as well as an ordinary repunit).

Below we list the record repunit primes with b > 2. The Mersennes (b=2) have their own pages.

### Record Primes of this Type

rankprime digitswhowhencomment
1(717624691 - 1)/7175 95202 CH2 Jun 2017 Generalized repunit
2(4073416111 - 1)/40733 74267 CH2 Mar 2015 Generalized repunit
3(6475815373 - 1)/64757 73960 p170 Jan 2018 Generalized repunit
4(5872915091 - 1)/58728 71962 CH2 Feb 2017 Generalized repunit
5(2798715313 - 1)/27986 68092 CH12 Aug 2020 Generalized repunit
6(2334015439 - 1)/23339 67435 p170 Mar 2020 Generalized repunit
7(2474115073 - 1)/24740 66218 p170 Jul 2020 Generalized repunit
8(6384713339 - 1)/63846 64091 p170 May 2013 Generalized repunit
9(2850713831 - 1)/28506 61612 CH12 Aug 2020 Generalized repunit
10(2637113681 - 1)/26370 60482 p170 Feb 2012 Generalized repunit
11(452916381 - 1)/4528 59886 CH2 Dec 2012 Generalized repunit
12(908215091 - 1)/9081 59729 CH2 Oct 2014 Generalized repunit
13(4332612041 - 1)/43325 55827 p170 Nov 2017 Generalized repunit
14(3828411491 - 1)/38283 52659 CH2 Feb 2013 Generalized repunit
15(3412011311 - 1)/34119 51269 CH2 Sep 2011 Generalized repunit
16(5009110357 - 1)/50090 48671 p170 Nov 2016 Generalized repunit
17(4449710093 - 1)/44496 46911 p170 Jun 2016 Generalized repunit
18(185213477 - 1)/1851 44035 p170 Aug 2015 Generalized repunit
19(424179337 - 1)/42416 43203 p170 Nov 2015 Generalized repunit
20(362109319 - 1)/36209 42480 p170 Mar 2019 Generalized repunit

### References

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