Primality tests for specific classes of $N=k\cdot 2^m \pm 1$

Explicit primality criteria for $h\cdot2^n\pm1$

We describe an explicit generalized Lucasian test to determine the primality of numbers $h\cdot2^n\pm1$ when $h\nequiv0\pmod{17}$. This test is by means of fixed seeds which depend only on $h$. In particular when $h=16^m-1$ with $m$ odd, our paper gives a primality test with some fixed seeds depending only on $h$. Comparing the results of W. Bosma(1993) and P. Berrizbeitia and T. G. Berry(2004), our result adds new values of $h$ along with this line. Octic and bioctic reciprocity are used to deduce our result.

A Modification of LLR

The Lucas-Lehmer (LL) primality test for Mersenne numbers is the fastest known primality test. In 1969, Hans Riesel published a modification of LL to test numbers of the form $N = h \cdot 2^n - 1$, where $h < 2^n$ is an odd integer and $n \ge 2$ \cite{Riesel}. This test is now known as the Lucas-Lehmer-Riesel (LLR) primality test. In Algorithm \ref{PrimalityAlgorithm}, we present a modification of LLR which works for any odd integer $N$. A probabilistic version of our algorithm runs in expected time $\tilde{O}(\log^3 N)$, and a deterministic version in expected $\tilde{O}(\log^4 N)$. We conclu…

不变量的定义及其与合数的关系：欧拉定理的推广

Definition of the invariant and the relationship with the compounds numbers. Generalisation of the Euler theorem

本文引入不变量的概念及其性质用于素性检验，结合欧拉定理将其推广至所有 $a^{φ(m)}$ 情形，并提出替代 Totient 函数以解释 Carmichael 数的新假设。

某些类广义弯曲函数的不存在性：重新审视元素划分方法

Nonexistence of certain classes of generalized bent functions: Revisiting the element partition method

本文利用元素划分方法，针对两类广义弯曲函数（类型 [n,q]）证明了新的不存在性结果，包括 q 为 2p1^{e1}p2^{e2} 和 q=2·3^a·7^b 的情形。

新的混合调和数与中心二项式系数的和

New sums mixing harmonic numbers and central binomial coefficients

本文提出了一种混合调和数与中心二项式系数的新和类，并建立了相关幂和的递归解。