Lucas non-Wieferich primes in arithmetic progressions and the <i>abc</i> conjecture

The abc conjecture and non-Wieferich primes in arithmetic progressions

Article history: Received 18 June 2012 Revised 12 September 2012 Accepted 6 October 2012 Available online xxxx Communicated by Greg Martin MSC: 11A41 11B25

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli.

Primes in arithmetic progressions

[1] Euler’s proof uses only simple properties of ζ(s), and only of ζ(s) as a function of a real, rather than complex, variable. Given the status of complex number and complex analysis in Euler’s time, this is not surprising. It is slightly more surprising that Dirichlet’s original argument also was a real-variable argument, since by that time, a hundred years later, complex analysis was well-es...

On primes in arithmetic progressions

Let d > 4 and c ∈ (−d, d) be relatively prime integers, and let r(d) be the product of all distinct prime divisors of d. We show that for any sufficiently large integer n (in particular n > 24310 suffices for 4 6 d 6 36) the least positive integer m with 2r(d)k(dk− c) (k = 1, . . . , n) pairwise distinct modulo m is just the first prime p ≡ c (mod d) with p > (2dn − c)/(d − 1). We also conjectu...

1 Arithmetic progressions in primes