Related papers: Kummer surfaces for primality testing
This paper presents two efficient primality tests that quickly and accurately test all integers up to $2^{64}$.
Generalized Cullen Numbers are positive integers of the form $C_b(n):=nb^n+1$. In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is…
In this paper, two approximation algorithms are given. Let N be an odd composite number. The algorithms give new directions regarding primality test of given N. The first algorithm is given using a new method called digital coding method.…
Odd numbers can be indexed by the map k(n)=(n-3)/2, n belonging to 2N+3. We first propose a basic primality test using this index function that was first introduced in article (8). Input size of operations is reduced which improves…
The purpose of this article is to delve into the properties of invariants. The properties, explained in [2], reveal new ways to develop algorithms that allow us to test the primality of a number. In this article, some of these are shown,…
The aim of this paper is to give an explicit extension of the classical elliptic integrals to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We study a family of Kummer surfaces corresponding to the Humbert surface of invariant $5$…
We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete…
In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as…
In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them…
For an elliptic curve with CM by $K$ defined over its Hilbert class field, $E/H$, we extend Lenstra's finite fields test to generators of norms of certain ideals in $\mathcal{O}_H$, yielding a sufficient $\widetilde{O}(\log^3 N)$ primality…
Integer counting processes increment of an integer value at transitions between states of an underlying Markov process. The generator of a counting process, which depends on a parameter conjugate to the increments, defines a complex…
A central question in mathematics and computer science is the question of determining whether a given ideal $I$ is prime, which geometrically corresponds to the zero set of $I$, denoted $Z(I)$, being irreducible. The case of principal…
We propose a multi-scale analysis method for studying arithmetic properties of integer sets, such as primality. Our approach organizes information through a hierarchy of nested sequences, where each level enables a hierarchical expression…
We present a quantum probabilistic algorithm which tests with a polynomial computational complexity whether a given composite number is of the Carmichael type. We also suggest a quantum algorithm which could verify a conjecture by…
We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin's unity, using the presentation of the Kummer…
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras called here…
Maximum likelihood iteration is one of the most commonly used reconstruction algorithms in quantum tomography. The main appeal of the method is that it is easy to implement and that it converges reliably to a physically meaningful density…
Many modern asymmetric encryption methods rely on prime numbers, as they have distinctive properties. For instance, the security of RSA cryptosystem relies on the computational difficulty of factoring a large composite number in its prime…
We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We…
We give an algorithm to decide whether an algebraic plane foliation F has a rational first integral and to compute it in the affirmative case. The algorithm runs whenever we assume the polyhedrality of the cone of curves of the surface…