A Brier number is a positive odd number that is both a Riesel number and a Sierpiński
number of the second kind. Equivalently, it is an odd
number 
such that 
and 
are composite
for every positive integer 
.
Known examples include 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 878503122374924101526292469, 3872639446526560168555701047, and 623506356601958507977841221247 (OEIS A076335). These have not been proved to be the smallest Brier numbers. Clavier (2014) gave finite lists of prime factors certifying the first example.
One generalization to any integer base 
requires a positive
integer 
to make 
and 
composite
for every positive integer 
, excluding the elementary fixed-divisor obstructions from
by the conditions 
, where 
is the greatest common
divisor. This generalized definition does not require 
to be an odd number. Virbalas
(2026, p. 18) proved that such generalized Brier numbers contain an infinite arithmetic progression for every base 
. In the construction, each of
and 
has at least two distinct prime
factors.
