Quadratic irrational integers with partly prescribed continued fraction expansion

We generalise remarks of Euler and of Perron by explaining how to detail all quadratic irrational integers for which the symmetric part of the period of their continued fraction expansion commences with prescribed partial quotients. The function field case is particularly striking.Comment: 10 pages; dedicated to the memory of Bela Brindz

Pseudo-elliptic integrals, units, and torsion

We remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomial whose square root generates a quadratic function field with non-trivial unit. We detail the genus 1 case.Comment: Submitted preprin

A problem around Mahler functions

Let $K$ be a field of characteristic zero and $k$ and $l$ be two multiplicatively independent positive integers. We prove the following result that was conjectured by Loxton and van der Poorten during the Eighties: a power series $F(z)\in K[[z]]$ satisfies both a $k$- and a $l$-Mahler type functional equation if and only if it is a rational function.Comment: 52 page

On the number of distinct prime factors of a sum of super-powers

Given k,ℓ∈N+, let xi,j be, for 1≤i≤k and 0≤j≤ℓ some fixed integers, and define, for every n∈N+, sn:=∑i=1 k∏j=0 ℓxi,j n. We prove that the following are equivalent: (a) There are a real θ>1 and infinitely many indices n for which the number of distinct prime factors of sn is greater than the super-logarithm of n to base θ.(b) There do not exist non-zero integers a0,b0,…,aℓ,bℓ such that s2n=∏i=0 ℓai (2n) and s2n−1=∏i=0 ℓbi (2n−1) for all n.We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes S, an effective bound on the number of non-degenerate solutions of an S-unit equation in k variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed c1,x1,…,ck,xk∈N+, the number of distinct prime factors of c1x1 n+⋯+ckxk n is bounded, as n ranges over N+, if and only if x1=⋯=xk