Related papers: A Computational Approach to Factoring Large Intege…

Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which has the requirement c>3/4. We introduce…

While it is trivial to multiply two C-finite sequences (just like integers), it is not quite so trivial to "factorize" them, or to decide whether they are "prime". The former is plain linear algebra, while the latter is heavy-duty…

We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and…

For an element $a$ of an integral domain D under an equivalence relation \tau, the \tau-factorization of a is defined as \lambda a_1 a_2... a_k, where \lambda is a unit in D and a_i \tau a_j for all i, j. An irreducible element has no…

We consider the problem of factoring permutations as a product of special types of transpositions, namely, those transpositions involving two positions with bounded distances. In particular, we investigate the minimum number, $\delta$, such…

Factorization is the most fundamental way to determine if a number $n$ is prime or composite. Yet, this approach becomes impracticable when considering large values of $n$, a difficulty that is exploited by cryptographic protocols. We…

We propose an alternative method to factorize an integer by using three harmonic oscillators. These oscillators are coupled together via specific Kerr nonlinear interactions. This method can be applied even if two harmonic oscillators are…

The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by seven or nine. For a positive non-prime integer N there is given an inductive way to find its…

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. We show that the smallest such $n$ is one of a pair of possibilities, or is one possibility, which we show how to calculate. There are three interesting numerical sequences which…

New algorithms for prime factorization that outperform the existing ones or take advantage of particular properties of the prime factors can have a practical impact on present implementations of cryptographic algorithms that rely on the…

We demonstrate a method for general linear optical networks that allows one to factorize any SU($n$) matrix in terms of two SU($n-1)$ blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in an…

While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct…

Approximating a definite integral of product of cosines to within an accuracy of n binary digits where the integrand depends on input integers x[k] given in binary radix, is equivalent to counting the number of equal-sum partitions of the…

For every sufficiently large integer $R$, there exists a Carmichael number with exactly $R$ prime factors.

We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model…

Quantum processors are potentially superior to their classical counterparts for many computational tasks including factorization. Circuit methods as well as adiabatic methods have already been proposed and implemented for finding the…

We extend a factorization due to Krein to arbitrary analytic functions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has…

We present a special-purpose algorithm for factoring semiprimes $N = pq$ in which one prime factor satisfies $p \approx c\,(a/b)^n$ for positive integers $a, b, c, n$ with $a > b$ and $\gcd(a,b) = 1$. Given the correct parameters $(a, b)$,…

Tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations -- where the input tensor and all the factors are required…

Factorization models express a statistical object of interest in terms of a collection of simpler objects. For example, a matrix or tensor can be expressed as a sum of rank-one components. However, in practice, it can be challenging to…