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# 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]:

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}

**Wagstaff prime**for primes of the form (2

*+1)/3 was first introduced by François Morain [Morain1990a]. The numbers (2*

^{p}*+1)/3 are probable primes for*

^{p}*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 (2^{127031}+ 1)/338240 E5 Jan 2023 Wagstaff, ECPP, generalized Lucas number 2 (2^{117239}+ 1)/335292 E2 Aug 2022 Wagstaff, ECPP, generalized Lucas number 3 (2^{95369}+ 1)/328709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP 4 (2^{83339}+ 1)/325088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff 5 (2^{42737}+ 1)/312865 M Aug 2007 ECPP, generalized Lucas number, Wagstaff 6 (2^{14479}+ 1)/34359 c4 Nov 2004 Generalized Lucas number, Wagstaff, ECPP 7 (2^{12391}+ 1)/33730 M Jun 1996 Generalized Lucas number, Wagstaff 8 (2^{11279}+ 1)/33395 PM Feb 1998 Cyclotomy, generalized Lucas number, Wagstaff 9 (2^{10691}+ 1)/33218 c4 Oct 2004 Generalized Lucas number, Wagstaff, ECPP 10 (2^{10501}+ 1)/33161 M May 1996 Generalized Lucas number, Wagstaff 11 (2^{5807}+ 1)/31748 PM Jan 1999 Cyclotomy, generalized Lucas number, Wagstaff 12 (2^{3539}+ 1)/31065 M Jan 1990 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 (2^{n}+1)/3 is prime from the On-Line Encyclopedia of Integer Sequences - Tony Reix's comments

### References

- 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.]

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