相关论文: A faster pseudo-primality test

In this paper, a random primality proving algorithm is proposed, which generates prime certificate of length O(log n). The certificate can be verified in deterministic time O(log^4 n). The algorithm runs in heuristical time tilde{O}(log^4…

A cyclic proof system allows us to perform inductive reasoning without explicit inductions. We propose a cyclic proof system for HFLN, which is a higher-order predicate logic with natural numbers and alternating fixed-points. Ours is the…

Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an…

In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1} \equiv 1 (\mathrm{mod} \ n)$ and $b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$ for all prime divisors $p$ of $n-1$ , then $n$ is prime, a result known as Lucas's…

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our approach is based on two components, each of which, we…

We present an alternative cyclic proof system for Peano arithmetic that could be simpler than the existing ones and well-adapted both for proof analysis and for automatizing inductive proof search. In addition, we will show how various…

Let $q$ be a power of a prime number $p$, $k=\mathbb{F}_{q}(t)$ be the rational function field over finite field $\mathbb{F}_{q}$ and $K/k$ be a multi-cyclic extension of prime degree. In this paper we will give an exact formula for the…

We study cyclic proof systems for $\mu\mathsf{PA}$, an extension of Peano arithmetic by positive inductive definitions that is arithmetically equivalent to the (impredicative) subsystem of second-order arithmetic $\Pi^1_2$-$\mathsf{CA}_0$…

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time…

Permutation tests are a popular choice for distinguishing distributions and testing independence, due to their exact, finite-sample control of false positives and their minimax optimality when paired with U-statistics. However, standard…

Let $\psi_m$ be the smallest strong pseudoprime to the first $m$ prime bases. This value is known for $1 \leq m \leq 11$. We extend this by finding $\psi_{12}$ and $\psi_{13}$. We also present an algorithm to find all integers $n\le B$ that…

The Birnbaum-Saunders regression model is commonly used in reliability studies. We address the issue of performing inference in this class of models when the number of observations is small. We show that the likelihood ratio test tends to…

A class of nonparametric two-sample tests has been proposed in this article. As a generalization of the original \v{S}id\'aks' test, the proposed test statistic is developed as the sum of the maximal precedence and maximal exceedance…

In this paper, we use the method of modified signed log-likelihood ratio test for the problem of testing the equality of correlation coefficients in two independent bivariate normal distributions. We compare this method with two other…

Possibilistic logic, an extension of first-order logic, deals with uncertainty that can be estimated in terms of possibility and necessity measures. Syntactically, this means that a first-order formula is equipped with a possibility degree…

We consider an application to the discrete log problem using completely regular semigroups which may provide a more secure symmetric cryptosystem than the classic system based on groups. In particular we describe a scheme that would appear…

We recently described a formalism for reasoning with if-then rules that re expressed with different levels of firmness [18]. The formalism interprets these rules as extreme conditional probability statements, specifying orders of magnitude…

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = \sigma + it$) is used to obtain estimates for $$…

For $k\ge1$, a $k$-almost prime is a positive integer with exactly $k$ prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of $k$-almost primes. Our results match…

This paper studies the problem of finding an $(1+\epsilon)$-approximate solution to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all…