On primary pseudo-polynomials (Around Ruzsa's Conjecture)

2021/02/02 by Eric, Delaygue, Tanguy, Rivoal
#11A41 #11B37 #11B50 (Primary) #33E20 #41A05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2102.01534

Abstract

Every polynomial P(X)∈ \mathbb Z[X] satisfies the congruences P(n+m)≡ P(n) \mod m for all integers n, m≥ 0. An integer valued sequence (a_n)_{n≥ 0} is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudo-polynomials and proved that they are not necessarily polynomials. A long standing conjecture of Ruzsa says that a pseudo-polynomial a_n is a polynomial as soon as \limsup_n \vert a_n\vert^1/n

Related

• On pseudo-polynomials divisible only by a sparse set of primes and \a-primary pseudo-polynomials
• An analogue of Ruzsa's conjecture for polynomials over finite fields
• On Ruzsa's conjecture on congruence preserving functions
• A note on an irreducible class of polynomials over integers
• New polynomials producing absolute pseudoprimes with any number of prime factors