Wagstaff

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

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 (2138937 + 1)/3 41824 E12 Oct 2023 Wagstaff, ECPP, generalized Lucas number 2 (2127031 + 1)/3 38240 E5 Jan 2023 Wagstaff, ECPP, generalized Lucas number 3 (2117239 + 1)/3 35292 E2 Aug 2022 Wagstaff, ECPP, generalized Lucas number 4 (295369 + 1)/3 28709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP 5 (283339 + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff 6 (242737 + 1)/3 12865 M Aug 2007 ECPP, generalized Lucas number, Wagstaff 7 (214479 + 1)/3 4359 c4 Nov 2004 Generalized Lucas number, Wagstaff, ECPP 8 (212391 + 1)/3 3730 M May 1996 Generalized Lucas number, Wagstaff 9 (211279 + 1)/3 3395 PM Jan 1998 Cyclotomy, generalized Lucas number, Wagstaff 10 (210691 + 1)/3 3218 c4 Oct 2004 Generalized Lucas number, Wagstaff, ECPP 11 (210501 + 1)/3 3161 M May 1996 Generalized Lucas number, Wagstaff 12 (25807 + 1)/3 1748 PM Dec 1998 Cyclotomy, generalized Lucas number, Wagstaff 13 (23539 + 1)/3 1065 M Dec 1989 First titanic by ECPP, generalized Lucas number, Wagstaff

Related Pages

• 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.]