# 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.

### 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:
The name Wagstaff prime for primes of the form (2p+1)/3 was first introduced by François Morain [Morain1990a]. The numbers (2p+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

rankprime digitswhowhencomment
1(2127031 + 1)/3 38240 E5 Jan 2023 Wagstaff, ECPP, generalized Lucas number
2(2117239 + 1)/3 35292 E2 Aug 2022 Wagstaff, ECPP, generalized Lucas number
3(295369 + 1)/3 28709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP
4(283339 + 1)/3 25088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff
5(242737 + 1)/3 12865 M Aug 2007 ECPP, generalized Lucas number, Wagstaff
6(214479 + 1)/3 4359 c4 Nov 2004 Generalized Lucas number, Wagstaff, ECPP
7(212391 + 1)/3 3730 M Jun 1996 Generalized Lucas number, Wagstaff
8(211279 + 1)/3 3395 PM Feb 1998 Cyclotomy, generalized Lucas number, Wagstaff
9(210691 + 1)/3 3218 c4 Oct 2004 Generalized Lucas number, Wagstaff, ECPP
10(210501 + 1)/3 3161 M May 1996 Generalized Lucas number, Wagstaff
11(25807 + 1)/3 1748 PM Jan 1999 Cyclotomy, generalized Lucas number, Wagstaff
12(23539 + 1)/3 1065 M Jan 1990 First titanic by ECPP, generalized Lucas number, Wagstaff

### 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 (2p+1)/3," J. Math. (Wuhan), 19 (1999) 199--202.  MR 2000i:11016 [The author proves the primality of (2p+1)/3 for p=1709 and 2617.]