Almost-primes represented by quadratic polynomials
We study the generalized random Fibonacci sequences defined by their first non-negative terms and for n≥1, Fn+2=λFn+1±Fn (linear case) and ̃Fn+2=|λ̃Fn+1±̃Fn| (non-linear case), where each ± sign is independent and either + with probability p or − with probability 1−p (0<p≤1). Our main result is that, when λ is of the form λk=2cos(π/k) for some integer k≥3, the exponential growth of Fn for 0<p≤1, and of ̃Fn for 1/k<p≤1, is almost surely positive and given by ∫0∞log x dνk, ρ(x),...
We provide a lower bound for the number of distinct zeros of a sum for two rational functions , in term of the degree of , which is sharp whenever have few distinct zeros and poles compared to their degree. This sharpens the “-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface contains only finitely many rational or elliptic curves,...
Zeta-generalized-Euler-constant functions, and defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and (1) = ln , are studied and estimated with high accuracy.