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Wagstaff |
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).
- p = 2k+/-1 or p = 4k+/-3
- 2p-1 is a prime (obviously a Mersenne prime)
- (2p+1)/3 is a prime.
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
- 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.]