The data on this page is associated with the paper "Consecutive primes which are widely digitally delicate and Brier numbers" by Michael Filaseta, Jacob Juillerat and Thomas Luckner. Note that the needed data from prior research, as indicated in the paper, can be found here. There are two new covering systems in the paper, one for establishing a number is Sierpinski and one for establishing a number is Riesel. For the purposes of the paper, these are interchangeable, though we indicate which is which below. For each covering system, there are two main lists to aid with following the constructions in the paper:
(i) a list of positive integers m and the primes p for which the order of 2 modulo p is m , and
(ii) a list that for the covering systems presented in tables in the paper.
Details regarding these lists are written below.
(i) a list of positive integers m and the primes p for which the order of 2 modulo p is m ,
Each element of this list is itself a list of the form [m,[p_1,...,p_t]]. The first entry m indicates that what follows is describing primes p for which the order of 2 modulo p is m. The m correspond to the moduli used in the specified covering system in the paper. In each case, p_1 < p_2 < ... < p_{t-1} and either p_{t-1} < p_{t} or p_{t-1} = p_{t}. If p_{t-1} < p_{t}, then the numbers p_1,...,p_{t-1} are distinct primes p for which the order of 2 modulo p is m. In this case, p_{t} is either a prime p for which the order of 2 modulo p is m or a composite number whose prime factors p are distinct from p_1,...,p_{t-1} and each prime factor p has the order of 2 modulo p equal to m. If p_{t-1} = p_{t}, then the numbers p_1,...,p_{t-2} are distinct primes p for which the order of 2 modulo p is m and p_{t-1} = p_{t} is a composite number which is not a power of a prime and which is relatively prime to the product of the primes p_1,...,p_{t-2}. Each prime divisor of p_{t} is such that the order of 2 modulo that prime is equal to m. These primes are determined by factoring or partially factoring Phi_{m}(2), where Phi_{m}(x) is the m-th cyclotomic polynomial. The list [m,[p_1,...,p_t]] is indicating that there are at least t primes for which 2 has order m modulo p .
(ii) a list that for the covering systems presented in tables in the paper.
This is a list of the form [a,m,p] corresponding to the congruence n = a (mod m) where p is a prime (or composite, but then p is associated with a prime divisor of this composite number) and m is the order of 2 modulo p. In other words, p appears in the list in (i) above for m. As above, p may appear twice (and hence m twice) if p is composite with at least two distinct prime factors. If q_1 and q_2 are two distinct prime factors of p in this case, then they are prime factors of Phi_{m}(2) which are relatively prime to m and hence 2 has order m modulo each of the primes q_1 and q_2. Thus, in the case that p is composite, the interpretation of [a,m,p] is that the congruence represented is n = a (mod m) and the prime p is one of q_1 and q_2, where each of these primes can be used with a different value of a.
Covering system/table for verifying the number is Sierpinski.
Covering system/table for verifying the number is Riesel.