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Prime rational functions

Omar Kihel, Jesse Larone (2015)

Acta Arithmetica

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.

Prime to p fundamental groups and tame Galois actions

Mark Kisin (2000)

Annales de l'institut Fourier

We show that for a local, discretely valued field F , with residue characteristic p , and a variety 𝒰 over F , the map ρ : Gal ( F sep / F ) → Out ( π 1 , geom ( p ' ) ( 𝒰 ) ) to the outer automorphisms of the prime to p geometric étale fundamental group of 𝒰 maps the wild inertia onto a finite image. We show that under favourable conditions ρ depends only on the reduction of 𝒰 modulo a power of the maximal ideal of F . The proofs make use of the theory of logarithmic schemes.

Primefree shifted Lucas sequences

Lenny Jones (2015)

Acta Arithmetica

We say a sequence = ( s ₙ ) n ≥ 0 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, a = ( u ₙ ) n ≥ 0 , 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 a ± k are simultaneously primefree. This result extends...

Primes in almost all short intervals. II

Danilo Bazzanella (2000)

Bollettino dell'Unione Matematica Italiana

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 g ⁢ n , g ⁢ n + H , con g ⁢ n 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 g ⁢ n , g ⁢ n + H con l'insieme eccezionale per la formula asintotica ψ ⁢ x + H - ψ ⁢ x ∼ H ⁢  as  ⁢ x → ∞ . I risultati presentati vengono quindi ottenuti grazie allo studio...

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