Related papers: Prime Factorization in Models of PV$_1$

There exists a Hamiltonian formulation of the factorisation problem which also needs the definition of a factorisation ensemble (a set to which factorable numbers, $N'=x'y'$, having the same trivial factorisation algorithmic complexity,…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha\in\mathbb{R}\backslash\mathbb{Q},\,\beta\in\mathbb{R}$ and $0<\theta<10/1561$, there…

Let $(X,\bullet )$ be a groupoid (binary algebra) and $Bin(X\dot{)}$ denote the collection of all groupoids defined on $X$. We introduce two methods of factorization for this binary system under the binary groupoid product \textquotedblleft…

Prime factorization (P = M*N) is considered to be a promising application in quantum computations. We perform 4-bit factorization in experiments using a superconducting flux qubit toward quantum annealing. Our proposed method uses a…

We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…

A notable result from analysis of Boolean functions is the Basic Invariance Principle (BIP), a quantitative nonlinear generalization of the Central Limit Theorem for multilinear polynomials. We present a generalization of the BIP for…

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…

We have presented a multivariate polynomial function termed as factor elimination function,by which, we can generate prime numbers. This function's mapping behavior can explain the irregularities in the occurrence of prime numbers on the…

We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a…

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the…

We investigate factorisation at small x using a variety of analytical and numerical techniques. Previous results on factorisation in collinear models are generalised to the case of the full BFKL equation, and illustrated in the example of a…

In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that…

This paper presents the first in a series of results that allow us to develop a theory providing finer control over the complexity of normalisation, and in particular of cut elimination. By considering atoms as self-dual non-commutative…

We develop a representative-level framework for the Liebscher-Tsirelson random-set construction of Arveson systems from stationary factorizing measure types. We introduce the notion of a measurable factorizing family of probability measures…

In this paper we give a factorization theorem for the ring of exponential polynomials in many variables over an algebraically closed field of characteristic 0 with an exponentiation. This is a generalization of the factorization theorem due…

The Frobenius number of relatively prime positive integers $a_1, \ldots, a_n$ is the largest integer that is not a nononegative integer combination of the $a_i.$ Given positive integers $a_1, \ldots, a_n$ with $n \ge 2,$ the set of…

Let $F$ be a field, $p$ a prime number, $X$ an indeterminate over $F$, $D_n =F[X^{\frac{1}{p^n}}, X^{-\frac{1}{p^n}}]$ for each integer $n \geq 0$ and $D = \bigcup\limits_{n\in\mathbb{N}_0}D_n.$ Then $D$ is a one-dimensional B{\'e}zout…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph…

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger…