The Top Twenty--a Prime Page Collection

Generalized Repunit

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The Prime Pages

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(288398317-1)/28838 37090 CH6 Dec 2006 Generalized repunit
2(130965953-1)/13095 24506 CH6 Nov 2007 Generalized repunit
3(231515347-1)/23150 23333 c13 Oct 2008 Generalized repunit
4(58556121-1)/5854 23058 CH1 Nov 2005 Generalized repunit
5(94734969-1)/9472 19756 CH2 Oct 2008 Generalized repunit
6(142614663-1)/14260 19367 c13 Nov 2007 Generalized repunit
7(137824591-1)/13781 19000 c13 Apr 2007 Generalized repunit
8(156374513-1)/15636 18925 c13 Nov 2007 Generalized repunit
9(180674201-1)/18066 17879 c13 Dec 2002 Generalized repunit
10(190264051-1)/19025 17332 c13 Mar 2008 Generalized repunit
11(47354621-1)/4734 16980 CH3 Sep 2005 Generalized repunit
12(110314177-1)/11030 16882 p54 Feb 2005 Generalized repunit
13(151343697-1)/15133 15450 CH6 Apr 2007 Generalized repunit
14(163393613-1)/16338 15219 c13 Mar 2008 Generalized repunit
15(40184177-1)/4017 15051 c13 Oct 2008 Generalized repunit
16(73723889-1)/7371 15038 CH6 Apr 2007 Generalized repunit
17(156793499-1)/15678 14676 x25 May 2003 Generalized repunit
18(27284231-1)/2727 14534 c13 Apr 2007 Generalized repunit
19(81853673-1)/8184 14369 c13 Jul 2003 Generalized repunit
20(38943967-1)/3893 14240 c13 Mar 2008 Generalized repunit

(up) Related Pages

(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, pp. xcvi+236, ISBN 0-8218-5078-4. 1988.  MR 90d:11009 (Annotation available)
Dubner93
H. Dubner, "Generalized repunit primes," Math. Comp., 61 (1993) 927--930.  MR 94a:11009
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
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., Boynton Beach, Florida, pp. vi+215, 1982.  MR 83k:10014
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