On the counting function of irregular primes

On the modularity of reducible mod $l$ Galois representations

“…The basic idea of using a 2 comes from the proof of a similar statement in weight 2 by Dieulefait, Jimenez Urroz and Ribet ([DJUR11, §2]). We are able to obtain a more general result because of our Theorem 1 (that generalizes Mazur's theorem to higher weight) and the information on primes p with large prime factors of p−1 given by Theorem 1 from [LMPM13]. It is conjectured that for any ε > 0, the set prime numbers p such that P + (p−1) ≥ p 1−ε has a positive lower density κ(ε) > 0.…”

“…It is conjectured that for any ε > 0, the set prime numbers p such that P + (p−1) ≥ p 1−ε has a positive lower density κ(ε) > 0. The bound κ(3/4) ≥ 3/4 is established in [LMPM13] by extending a method of Goldfeld who had previously obtained κ(1/2) ≥ 1/2 ([Gol69]). Much effort has been invested in solving this conjecture for values of ε as small as possible (cf.…”

“…As a new ingredient, we use results from analytic number theory on the density of prime numbers p for which p − 1 has a large prime factor (cf. [Gol69], [LMPM13]). We then manage to obtain a bound which is better than (0.2) but is only valid in a restricted class of prime numbers.…”

Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture

“…Let P B be the set of B-irregular primes. Carlitz [3] gave a simple proof of the infinitude of this set, and recently Luca, Pizarro-Madariaga and Pomerance [20,Theorem 1] made this more quantitative by showing that…”

Denominators of Bernoulli Polynomials

“…Finally, we use a recent improvement due to McNew, Pollack and Pomerance [17] of a result of Erdős and Wagstaff [6] on the count of positive integers n divisible by shifted primes (see also [7]), as well as a result of Luca, Pizarro-Madariaga and Pomerance [15] about shifted primes divisible by another shifted prime.…”