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. Definition: 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 Questions: 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?

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On October 20, 2025, Wilfrid Keller wrote:

Concerning (b):

To my knowledge, Christophe Clavier's December 2013 record of the smallest known Brier number (22 digits long) remained unbroken ever since. 

 

Concerning (c):
I am pleased to report some happy news that emerged this month (October 6, 2025) when Kellen Shenton discovered the final prime 38118498221*2^7552807+1 (with more than two million digits!) needed to complete the proof that "There are no Brier numbers below 10^11". (Please compare  https://t5k.org/primes/page.php?id=141134 ).

 
The completion of the previous step 10^10 had been announced at OEIS five years earlier  (October 25, 2020), where "the largest necessary prime" was identified as 1355477231*2^356981+1.
 
If I may mention it (in all modesty), I had submitted that prime to the PrimePages already in April 2002 ( https://t5k.org/primes/page.php?id=107 ), completing the search for k < 2^31 = 2147483648. Actually, I can provide the results of an independent "proof" for k < 10^10 by showing the smallest prime p(k) obtained for every k whenever p(k) > 10000 (570 cases). [Reference to file Proof-10.txt]
 
If you are interested in the chronological development of the search, it is remarkable to note that Kellen Shenton had almost finished 10^11 (with only one exception remaining!) as early as in May 2021, finally getting the above "38118498221*2^7552807+1 is prime" in this month.  
 
In the course of this work, Kellen Shenton was able to supply three new entries for the "stepladder". See the update to Problem 30.

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