Problems & Puzzles: Puzzles

Problem 87. More 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 A076335:

3316923598096294713661(22 digits), 10439679896374780276373 (prime, 23 digits), 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, 21444598169181578466233 (prime, 23 digits), 28960674973436106391349, 32099522445515872473461, 32904995562220857573541(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 sma
ller 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?

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