A Wagstaff prime is a prime number of the form 
for 
an odd prime number. The following
list gives the initial exponents of known primes
or probable primes as of Oct. 2026. The values
are 
,
5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701,
1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239,
127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ... (OEIS A000978),
with 
and larger corresponding to probable primes. These
values 
correspond to the primes 
with indices 
, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, ... (OEIS A123176).
The Wagstaff primes are featured in the new Mersenne prime conjecture.
Known tests analogous to the Lucas-Lehmer test for Wagstaff primes remain conjectural as sufficient
primality tests (Dolotov 2026, p. 1). Morain
(1991) used elliptic curve primality
proving to prove that 
is prime.
A Wagstaff prime can also be interpreted as a repunit prime of base 
,
as
 |
if 
is odd, as it must be for the above number to be prime.
The following table summarizes some Wagstaff probable primes reported through 2021. In Sep. 2013, the terms with 
and 
were the largest two known probable
primes (Propper 2013, Lifchitz and Lifchitz), but were not necessarily the sequentially
next primes after 
.
 | decimal digits | discoverer |
| 374321 |  | H. R. Lifchitz (Dec. 2000) |
| 986191 |  | V. Diepeveen (Jun. 2008) |
| 4031399 |  | T. Reix et al. (Feb. 2010) |
| 13347311 |  | R. Propper (Sep. 2013) |
| 13372531 |  | R. Propper (Sep. 2013) |
| 15135397 |  | R. Propper (reported 2021) |
A 2021 report credits R. Propper with finding the Wagstaff probable prime 
(OEIS A000978). It has 4,556,209 decimal
digits.
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