Primality tests, linear recurrent sequences and the Pell equation

基于佩尔三次曲线的新型高效素性测试

Novel performant primality test on a Pell's cubic

本研究提出一种基于佩尔三次曲线的新型素性测试算法，通过相关整数序列证明其必要素性条件，运算量随输入整数比特长度线性增长，对小于2的32次方的整数具有确定性，为公钥密码学提供了新的素性测试方案。

New properties for a composition of some generating functions for primes

In this paper, we consider properties of coefficients of a generating functions composition, where the outer function is a logarithmic generating function and the inner function is an ordinary generating function with integer coefficients. Using notions of composita and composition of generating functions, we get new properties for this composition. The properties can be used for distinguishing prime numbers from composite numbers. As an application, obtained results can be used to obtain new primality criteria. We obtain primality criteria for the Mersenne numbers, the Lucas numbers, the Pell…

Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations

For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell equations. With some conditions on parities of fundamental solutions of the simple and generalized Pell equations, only odd solutions of the generalized Pell equation are retained to provide many infinitely solutions found on branches corresponding to each of the generalized fundamental solutions. General algebraic expressions of fundamental solutions of the P…

On rational approximations of algebraic numbers of higher orders and certain parametrization for generalized Pell's equations

A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell's equations is constructed.

On symmetric powers of $τ$-recurrent sequences and deformations of Eisenstein series

We prove the equality of several $τ$-recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an $A$-expansion for the $l^\text{th}$ power ($1 \leq l \leq q$) of the deformation of the weight 2 Eisenstein series; relations between Drinfeld modular forms with $A$-expansions; a new proof of relations between special values of Pellarin $L$-series.

Diophantine Equation with Arithmetic functions and Binary recurrent sequences

This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form $φ(|au_n |)=|bv_m|,$ where $φ$ is the Euler totient function, $\{u_n\}_{n\geq 0}$ and $\{v_m\}_{m\geq 0}$ ar…