Let (u n ) n 0 be a nondegenerate Lucas sequence and g u (n) be the arithmetic function defined by gcd(n, u n ). Recent studies have investigated the distributional characteristics of g u . Numerous results have been proven based on the two extreme values 1 and n of g u (n). Sanna investigated the average behaviour of g u and found asymptotic formulas for the moments of log g u . In a related direction, Jha and Sanna investigated properties of g u at shifted primes.In light of these results, we prove that for each positive integer λ , we havewhere P u,λ is a constant depending on u and λ which is expressible as an infinite series. Additionally, we provide estimates for P u,λ and M u,λ , where M u,λ is the constant for an analogous sum obtained by Sanna [J. Number Theory 191 (2018), 305-315]. As an application of our results, we prove upper bounds on the count #{p x : g u (p − 1) > y} and also establish the existence of infinitely many runs of m consecutive primes p in bounded intervals such that g u (p − 1) > y based on a breakthrough of Zhang, Maynard, Tao, et al. on small gaps between primes. Exploring further in this direction, it turns out that for Lucas sequences with nonunit discriminant, we have max{g u (n) : n x} ≫ x. As an analogue, we obtain that that max{g u (p − 1) : p x} ≫ x 0.4736 unconditionally, while max{g u (p − 1) : p x} ≫ x 1−o(1) under the hypothesis of Montgomery's or Chowla's conjecture.

Cilleruelo conjectured that if [Formula: see text] is an irreducible polynomial of degree [Formula: see text] then, [Formula: see text] In this paper, we investigate the analog of prime arguments, namely, [Formula: see text] where [Formula: see text] denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of [Formula: see text]

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Prediction of carbon monoxide (CO) emission is critical in gas turbine combustion. Compact yet accurate reaction mechanisms are required to predict CO with reasonable computing cost. This study uses SHERPA optimization algorithm to optimize the kinetic rate parameters of a 3-step methane-air global reaction mechanism for improved CO predictions. DARS is used as the chemical kinetics solver. Freely propagating laminar flame and constant pressure reactor solutions with GRI-Mech 3.0 reaction mechanism are used as references for optimization. Tradeoffs in the choice of solution techniques and solver settings for fast and accurate design runs are discussed in the paper. Optimization results and their interpretation for improving the design study is also presented. The optimal results show significant improvements in predictions compared to the baseline case. The workflow and best practices presented in this paper may be extended to optimize global reaction mechanisms for any given range of operating conditions.

We investigate, for given positive integers a and b, the least positive integer $c=c(a,b)$ such that the quotient $\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$ is an integer. We derive results on the limit of $c(a,b)/(a+b)$ as a and b tend to infinity and show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ , with the exception of a set of density zero.