Related papers: A geometric approach to integer factorization
The basic elements of the geometric approach to a consistent quantization formalism are summarized, with reference to the methods of the old quantum mechanics and the induced representations theory of Lie groups. A possible relationship…
In this note we describe a new method of counting the number of unordered factorizations of a natural number by means of a generating function and a recurrence relation arising from it, which improves an earlier result in this direction.
Integer factorization is a famous computational problem unknown whether being solvable in the polynomial time. With the rise of deep neural networks, it is interesting whether they can facilitate faster factorization. We present an approach…
In this article, we considered a fractal image as a fractal curve, that is, as a walk on a grid in Euclidean space $\R^d$. We placed integers on the generating vectors of a grid, such that opposite directions have opposite numbers. This…
We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…
Let $n = \mathrm{p}\!\cdot\!\mathrm{q}$ (p < q) and $\Delta = \lvert p-q \rvert$, where p,q are odd integers, then, it is hypothesized that factorizing this composite n will take O(1) time once the steady state value is reached for any…
We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based…
We apply the geometric construction of solutions of some variational problems of combinatorics to estimate the number of partitions and of plane partitions of an integer.
Expectation propagation is a general approach to fast approximate inference for graphical models. The existing literature treats models separately when it comes to deriving and coding expectation propagation inference algorithms. This comes…
An elementary, albeit higher dimensional, argument is used to compute the area under the power function curve between 0 and 1.
In this paper, we create a systematic and automatic procedure for transforming the integer factorization problem into the problem of solving a system of Boolean equations. Surprisingly, the resulting system of Boolean equations takes on a…
We study the integer points on superelliptic and hyperelliptic curves of the form $y^n=f(x)g(x),$ $n\ge 2, {\rm{deg}}{f}+{\rm{deg}}{g}\ge 4.$
In this paper, we study numbers $n$ that can be factored in three different ways as $n = A_1 B_1 = A_2 B_2 = A_3 B_3$ where $A_1$, $A_2$, $A_3$ are close to each other and $B_1$, $B_2$, $B_3$ are close to each other.
A matrix factorization problem is considered. The matrix to be factorized is algebraic, has dimension 2 X 2 and belongs to Moiseev's class. A new method of factorization is proposed. First, the matrix factorization problem is reduced to a…
We describe a novel analogue algorithm that allows the simultaneous factorization of an exponential number of large integers with a polynomial number of experimental runs. It is the interference-induced periodicity of "factoring"…
An integer $n$ is $(y,z)$-semismooth if $n=pm$ where $m$ is an integer with all prime divisors $\le y$ and $p$ is 1 or a prime $\le z$. arge quantities of semismooth integers are utilized in modern integer factoring algorithms, such as the…
We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the…
In this note, we represent integers in a type of factoradic notation. Rather than use the corresponding Lehmer code, we will view integers as permutations. Given a pair of integers n and k, we give a formula for n mod k in terms of the…
In the present paper we provide a probabilistic polynomial time algorithm that reduces the complete factorization of any squarefree integer $n$ to counting points on elliptic curves modulo $n$, succeeding with probability $1-\varepsilon$,…
In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function…