Wagstaff

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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.

Definitions and Notes

Bateman, Selfridge, and Wagstaff have made the The New Mersenne Conjecture [BSW89]:

Let p be any odd natural number. If two of the following conditions hold, then so does the third:

p = 2^k+/-1 or p = 4^k+/-3

2^p-1 is a prime (obviously a Mersenne prime)

(2^p+1)/3 is a prime.

The name Wagstaff prime for primes of the form (2^p+1)/3 was first introduced by François Morain [Morain1990a]. The numbers (2^p+1)/3 are probable primes for p = 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191 (Diepeveen 2008), 4031399 (Vrba, Reix 2010); also 13347311 and 13372531 (Ryan 2013).

Record Primes of this Type

rank prime digits who when comment

1 (2⁹⁵³⁶⁹ + 1)/3 28709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP
2 (2⁸³³³⁹ + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff
3 (2⁴²⁷³⁷ + 1)/3 12865 M Aug 2007 ECPP, generalized Lucas number, Wagstaff
4 (2¹⁴⁴⁷⁹ + 1)/3 4359 c4 Nov 2004 Generalized Lucas number, Wagstaff, ECPP
5 (2¹²³⁹¹ + 1)/3 3730 M May 1996 Generalized Lucas number, Wagstaff
6 (2¹¹²⁷⁹ + 1)/3 3395 PM Jan 1998 Cyclotomy, generalized Lucas number, Wagstaff
7 (2¹⁰⁶⁹¹ + 1)/3 3218 c4 Oct 2004 Generalized Lucas number, Wagstaff, ECPP
8 (2¹⁰⁵⁰¹ + 1)/3 3161 M May 1996 Generalized Lucas number, Wagstaff
9 (2⁵⁸⁰⁷ + 1)/3 1748 PM Dec 1998 Cyclotomy, generalized Lucas number, Wagstaff
10 (2³⁵³⁹ + 1)/3 1065 M Dec 1989 First titanic by ECPP, generalized Lucas number, Wagstaff

Status of the New Mersenne Prime Conjecture Originally by Conrad Curry
Status of the New Mersenne Prime Conjecture by Renaud Lifchitz
Numbers n such that (2ⁿ+1)/3 is prime from the On-Line Encyclopedia of Integer Sequences
Tony Reix's comments

References

BSW89
P. T. Bateman, J. L. Selfridge and Wagstaff, Jr., S. S., "The new Mersenne conjecture," Amer. Math. Monthly, 96 (1989) 125-128. MR 90c:11009
LRS1999
Leyendekkers, J. V., Rybak, J. M. and Shannon, A. G., "An analysis of Mersenne-Fibonacci and Mersenne-Lucas primes," Notes Number Theory Discrete Math., 5:1 (1999) 1--26. MR 1738744
Morain1990a
F. Morain, Distributed primality proving and the primality of (2³⁵³⁹+1)/3. In "Advances in cryptology---EUROCRYPT '90 (Aarhus, 1990)," Lecture Notes in Comput. Sci. Vol, 473, Springer, 1991. Berlin, pp. 110--123, MR1102475
Pi1999
X. M. Pi, "Primes of the form (2^p+1)/3," J. Math. (Wuhan), 19 (1999) 199--202. MR 2000i:11016 [The author proves the primality of (2^p+1)/3 for p=1709 and 2617.]