Given positive integers a1,…,aka1,…,ak, we prove that the set of primes p such that p≢1 mod ai for i=1,…,k admits asymptotic density relative to the set of all primes which is at least ∏ki=1(1−1/φ(ai), where φ is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer n such that n≢0mod ai for i=1,…,ki=1,…,k admits asymptotic density which is at least ∏ki=1(1−1/ai).

A note on primes in certain residue classes / Leonetti, Paolo; Sanna, Carlo. - In: INTERNATIONAL JOURNAL OF NUMBER THEORY. - ISSN 1793-0421. - STAMPA. - 14:8(2018), pp. 2219-2223. [10.1142/S1793042118501336]

A note on primes in certain residue classes

Sanna, Carlo
2018

Abstract

Given positive integers a1,…,aka1,…,ak, we prove that the set of primes p such that p≢1 mod ai for i=1,…,k admits asymptotic density relative to the set of all primes which is at least ∏ki=1(1−1/φ(ai), where φ is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer n such that n≢0mod ai for i=1,…,ki=1,…,k admits asymptotic density which is at least ∏ki=1(1−1/ai).
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11583/2722602