Let f(x) be a complex rational function. We study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we also derive some conditions for the case of complex polynomials.
We show that for a local, discretely valued field , with residue characteristic , and a variety over , the map to the outer automorphisms of the prime to geometric étale fundamental group of maps the wild inertia onto a finite image. We show that under favourable conditions depends only on the reduction of modulo a power of the maximal ideal of . The proofs make use of the theory of logarithmic schemes.
We say a sequence is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-2 for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences are simultaneously primefree. This result extends...
In questo lavoro vengono migliorati i risultati ottenuti in «Primes in Almost All Short Intervals» riguardo la distribuzione dei primi in quasi tutti gli intervalli corti della forma , con funzione reale appartenente ad una ampia classe di funzioni. Il problema viene trattato mettendo in relazione l'insieme eccezionale per la distribuzione dei primi in intervalli nella forma con l'insieme eccezionale per la formula asintotica I risultati presentati vengono quindi ottenuti grazie allo studio...