Problems & Puzzles:
questions about Brier integers
On August 8, 2023 Carlos Rivera posted the following problem
Brier numbers is an issue that we have dealt with before in
several problems: 29, 49, 52, 58, 68 & 69.
A Brier number is an integer k such that k.2^n+1 and k.2^n-1
are composites for all n. Other way to define it is to say
that A Brier number is an integer k that is at the same time
a Sierpinski and a Riesel integer.
Apparently Eric Brier discovered the first one of such kind
on integers on 28/9/98:
k=29364695660123543278115025405114452910889 (41 digits,
composite). But later, Arkadiusz Wesolowski reported that
the very first one known "Brier integer" was discovered by
Fred Cohen and J. L. Selfridge in 1975:
k=47867742232066880047611079 (26 digits, prime)
Up to this date, the 9 smallest Brier integers known are in
3316923598096294713661(22 digits), 10439679896374780276373
(prime, 23 digits), 11615103277955704975673,
21444598169181578466233 (prime, 23 digits),
Of these nine Brier smaller integers, only
10439679896374780276373 & 21444598169181578466233 are prime
numbers. All the other are composite integers.
On another direction, on Oct 25, 2020 Kellen Shenton
reported that there are not Brier integers les than 10^10
a) Find a smaller prime Brier interger than
10439679896374780276373 (23 digits, prime) gotten by Dan
Ismailescu & P. S. Park in 2013.
b) Find a smaller Brier integer than 3316923598096294713661
(22 digits, composite) goten by Christopher Clavier in 2014.
c) Can you extend the range given by K. Shenton in 2020?