The Top Twenty--a Prime Page Collection

Generalized Repunit

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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. Comments and suggestions requested.

(up) 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: \sigma (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.

(up) Record Primes of this Type

rankprime digitswhowhencomment
1(4073416111 - 1)/40733 74267 CH2 Mar 2015 Generalized repunit
2(6384713339 - 1)/63846 64091 p170 May 2013 Generalized repunit
3(2637113681 - 1)/26370 60482 p170 Feb 2012 Generalized repunit
4(452916381 - 1)/4528 59886 CH2 Dec 2012 Generalized repunit
5(908215091 - 1)/9081 59729 CH2 Oct 2014 Generalized repunit
6(3828411491 - 1)/38283 52659 CH2 Feb 2013 Generalized repunit
7(3412011311 - 1)/34119 51269 CH2 Sep 2011 Generalized repunit
8(4449710093 - 1)/44496 46911 p170 Jun 2016 Generalized repunit
9(185213477 - 1)/1851 44035 p170 Aug 2015 Generalized repunit
10(424179337 - 1)/42416 43203 p170 Nov 2015 Generalized repunit
11(325569283 - 1)/32555 41887 CH2 Apr 2011 Generalized repunit
12(154912973 - 1)/1548 41382 p170 Dec 2010 Generalized repunit
13(131179151 - 1)/13116 37679 CH2 Jan 2016 Generalized repunit
14(288398317 - 1)/28838 37090 CH6 Dec 2006 Generalized repunit
15(436610099 - 1)/4365 36758 x14 Mar 2011 Generalized repunit
16(200608377 - 1)/20059 36037 c67 Apr 2015 Generalized repunit
17(113797411 - 1)/11378 30056 x14 Dec 2009 Generalized repunit
18(133206997 - 1)/13319 28856 x14 Oct 2010 Generalized repunit
19(34297549 - 1)/3428 26684 x14 Jul 2009 Generalized repunit
20(130965953 - 1)/13095 24506 CH6 Nov 2007 Generalized repunit

(up) References

AG1974
I. O. Angell and H. J. Godwin, "Some factorizations of 10n± 1," Math. Comp., 28 (1974) 307--308.  MR 48:8366
Beiler1964
A. Beiler, Recreations in the theory of numbers, Dover Pub., New York, NY, 1964.
BLSTW88
J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff, Jr., Factorizations of bn ± 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., Providence RI, 1988.  pp. xcvi+236, ISBN 0-8218-5078-4. MR 90d:11009 (Annotation available)
Dubner2002
Dubner, Harvey, "Repunit R49081 is a probable prime," Math. Comp., 71:238 (2002) 833--835 (electronic).  (http://dx.doi.org/10.1090/S0025-5718-01-01319-9) MR 1885632 (Abstract available)
Dubner93
H. Dubner, "Generalized repunit primes," Math. Comp., 61 (1993) 927--930.  MR 94a:11009
Jaroma2009
Jaroma, John H., On primes and pseudoprimes in the generalized repunitsCongr. Numer., In "Proceedings of the Fortieth Southeastern International Conference on Combinatorics, Graph Theory and Computing," Vol, 195, 2009.  pp. 105--114, MR 2584289
Oblath1956
R. Obláth, "Une propriété des puissances parfaites," Mathesis, 65 (1956) 356--364.
Rotkiewicz1987
A. Rotkiewicz, "Note on the diophantine equation 1 + x + x2 + ... + xn = ym," Elem. Math., 42:3 (1987) 76.  MR 88c:11020
Salas2011
Salas, Christian, "Base-3 repunit primes and the Cantor set," Gen. Math., 19:2 (2011) 103--107.  MR 2818401
WD86
H. C. Williams and H. Dubner, "The primality of R1031," Math. Comp., 47:176 (1986) 703--711.  MR 87k:11141
Yates82
S. Yates, Repunits and repetends, Star Publishing Co., Inc., 1982.  Boynton Beach, Florida, pp. vi+215, MR 83k:10014
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