Related papers: Positive spoof Lehmer factorizations
The starting point of this paper are the Mittag-Leffler polynomials introduced by H. Bateman [1]. Based on generalized integer powers of real numbers and deformed exponential function, we introduce deformed Mittag-Leffler polynomials…
In this paper we present a new version of the second author's factorization theorem for perfect matchings of symmetric graphs. We then use our result to solve four open problems of Propp on the enumeration of trimer tilings on the hexagonal…
Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative…
Diophantine quadruples are sets of four distinct positive integers such that the product of any two is one less than a square. All known examples belong to an infinite set which can be constructed recursively. Some observations on these…
The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series, is nowadays beyond dispute. Weniger's…
The elementary theory of bivariate linear Diophantine equations over polynomial rings is used to construct causal lifting factorizations (elementary matrix decompositions) for causal two-channel FIR perfect reconstruction transfer matrices…
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be…
We consider Diophantine equations of the kind $|F(x,y)|= m$, where $F(X,Y )\in \bz [X,Y]$ is a homogeneous polynomial of degree $d\ge 3$ that has non-zero discriminant and $m$ is a positive integer. We prove results that simplify those of…
A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm…
Probabilistic Latent Tensor Factorization (PLTF) is a recently proposed probabilistic framework for modelling multi-way data. Not only the common tensor factorization models but also any arbitrary tensor factorization structure can be…
We propose a splitting algorithm for solving a system of composite monotone inclusions formulated in the form of the extended set of solutions in real Hilbert spaces. The resluting algorithm is a an extension of the algorithm in [4]. The…
We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the…
We describe three algorithms for computer-aided symbolic multi-loop calculations that facilitated some recent novel results. First, we discuss an algorithm to derive the canonical form of an arbitrary Feynman integral in order to facilitate…
It has been well known since Fermat's Little Theorem was first published that it is possible to determine that a number is composite without determining any of its nontrivial factors. It is natural to ask whether it is also possible to…
By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^2=f(x)^2 \pm f(y)^2$ for some simple Laurent polynomials $f$.
In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…
For K \subseteq C, let B_n(K)={(x_1,...,x_n) \in K^n: for each y_1,...,y_n \in K the conjunction (\forall i \in {1,...,n} (x_i=1 => y_i=1)) AND (\forall i,j,k \in {1,...,n} (x_i+x_j=x_k => y_i+y_j=y_k)) AND (\forall i,j,k \in {1,...,n}…
Consider a sparse polynomial in several variables given explicitly as a sum of non-zero terms with coefficients in an effective field. In this paper, we present several algorithms for factoring such polynomials and related tasks (such as…