## 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. Comments and suggestions requested.

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

LetThe namepbe any odd natural number. If two of the following conditions hold, then so does the third:

p= 2+/-1 or^{k}p= 4+/-3^{k}- 2
-1 is a prime (obviously a Mersenne prime)^{p}- (2
+1)/3 is a prime.^{p}

rank prime digits who when comment 1 (2^{95369}+ 1)/328709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP 2 (2^{83339}+ 1)/325088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff 3 (2^{42737}+ 1)/312865 M Aug 2007 ECPP, generalized Lucas number, Wagstaff 4 (2^{14479}+ 1)/34359 c4 Nov 2004 Generalized Lucas number, Wagstaff, ECPP 5 (2^{12391}+ 1)/33730 M May 1996 Generalized Lucas number, Wagstaff 6 (2^{11279}+ 1)/33395 PM Jan 1998 Cyclotomy, generalized Lucas number, Wagstaff 7 (2^{10691}+ 1)/33218 c4 Oct 2004 Generalized Lucas number, Wagstaff, ECPP 8 (2^{10501}+ 1)/33161 M May 1996 Generalized Lucas number, Wagstaff 9 (2^{5807}+ 1)/31748 PM Dec 1998 Cyclotomy, generalized Lucas number, Wagstaff 10 (2^{3539}+ 1)/31065 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 (2^{n}+1)/3 is prime from the On-Line Encyclopedia of Integer Sequences - Tony Reix's comments

- BSW89
P. T. Bateman,J. L. SelfridgeandWagstaff, Jr., S. S., "The new Mersenne conjecture,"Amer. Math. Monthly,96(1989) 125-128.MR 90c:11009- LRS1999
Leyendekkers, J. V.,Rybak, J. M.andShannon, 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 (2. In "Advances in cryptology---EUROCRYPT '90 (Aarhus, 1990)," Lecture Notes in Comput. Sci. Vol, 473, Springer, 1991. Berlin, pp. 110--123,^{3539}+1)/3MR1102475- 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 forp=1709 and 2617.]

Chris K. Caldwell
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