Narrow Results

This study investigates numbers that are powers of two and can be expressed as the sum of the squares of any two Pell numbers. We apply Baker’s theory of linear forms in logarithms of algebraic numbers, along with a variation of the Baker–Davenport reduction method, to solve the Diophantine equation $P_m^2+P_n^2=2^a$, where $m,n$, and $a$ are non-negative integers, as presented here. Furthermore, the maple${}^{\text{©}}$ codes used in the computations made throughout the paper are also shared.

We give a brief survey of results that use “reversal-bounded counters” in studying reachability problems for various classes of transition systems. We also discuss the connection between the decidability of reachability in counter machines and the solvability of certain Diophantine equations.

Using a directed acyclic graph (dag) model of algorithms, we solve a problem related to precedence-constrained multiprocessor schedules for array computations: Given a sequence of dags and linear schedules parametrized by n, compute a lower bound on the number of processors required by the schedule as a function of n. In our formulation, the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions d_n to a set of parametric linear Diophantine equations. We present an algorithm based on generating functions for constructing a formula for these numbers d_n. The algorithm has been implemented as a Mathematica program. Example runs and the symbolic formulas for processor lower bounds automatically produced by the algorithm for Matrix-Vector Product, Triangular Matrix Product, and Gaussian Elimination problems are presented. Our approach actually solves the following more general problem: Given an arbitrary r× s integral matrix A and r-dimensional integral vectors b and c, let d_n(n=0,1,…) be the number of solutions in non-negative integers to the system Az=nb+c. Calculate the (rational) generating function ∑_{n≥ 0}d_ntⁿ and construct a formula for d_n.

We give an overview of probabilistic phenomena in metric discrepancy theory and present several new results concerning the asymptotic behavior of discrepancies D_N(n_kx) and sums ∑_ckf(n_kx) for sequences (n_k)_{k ≥ 1} of integers.

While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ₂(Z) which is not clean.

In this paper, it is proved that the Diophantine equation x⁴-y⁴ =z² has no non-trivial coprime solutions in the rings of integers of quadratic imaginary fields for d=11, 19, 43, 67, 163, which implies that the Fermat equation x⁴+y⁴ =z⁴ has no non-trivial solutions in these fields either. Then all the solutions of the Pocklington equation x⁴-x²y²+y⁴ =(-1)^σz² (σ =0 or 1) in the ring of integers of are determined, and as an application, the result is applied to K₂ of a field.

In this note, we find all the solutions of the Diophantine equation x² + 2^a · 5^b = yⁿ in positive integers x, y, a, b, n with x and y coprime and n ≥ 3.

It is well known that the Diophantine equations x⁴ + y⁴ = z⁴ + w⁴ and x⁴ + y⁴ + z⁴ = w⁴ each have infinitely many rational solutions. It is also known for the equation x⁶ + y⁶ - z⁶ = w². We extend the investigation to equations x^a ± y^b = ±z^c ± w^d, a, b, c, d ∈ Z, with 1/a + 1/b + 1/c + 1/d = 1. We show, with one possible exception, that if there is a solution of the equation in the reals, then the equation has infinitely many solutions in the integers. Of particular interest is the equation x⁶ + y⁶ + z⁶ = w² because of its classical nature; but there seem to be no references in the literature.

Let a, b, c be integers. In this paper, we prove the integer solutions of the equation ax^y + by^z + cz^x = 0 satisfy max{|x|, |y|, |z|} ≤ 2 max{a, b, c} when a, b, c are odd positive integers, and when a = b = 1, c = -1, the positive integer solutions of the equation satisfy max{x, y, z} < exp(exp(exp(5))).

We investigate the divisors $d$ of the numbers $P(n)$ for various polynomials $P\in \mathbb{Z}[x]$ such that $d\equiv 1(\mathrm{\text{mod}}n)$. We obtain the complete classification of such divisors for a class of polynomials, in particular for $P(x)=x^4+1$. We also construct a fast algorithm which provides all such factorizations up to a given limit for another class, for example for $P(x)=2x^4+1$. We use these results to find all the divisors $d=2^{m}k+1$ of numbers $2^{4m}+1$ and $2^{4m+1}+1$. For the numbers $2^{4m}+1$ the complete classification of such divisors is provided while for the numbers $2^{4m+1}+1$ the given classification is proved to be exhaustive only for $m\le 1000$.

In this paper, we study the Diophantine equation ${(x-d)}^4+x^4+{(x+d)}^4=y^n$, and based on Frey–Hellegouarch curves and their corresponding Galois representations, we solve the equation for various choices of $d$.

The purpose of the present paper is to use the improvement of Runge’s theorem on diophantine equations due to Schinzel to prove that if $P$ has at least three different roots then the equation in the title has a finite number of solutions.

We show that integral solutions of $x^3+y^3+z^3=q$, where $q$ is prime, satisfy certain congruence conditions.

Let $[𝜃]$ denote the integral part of the real number $𝜃$. In this paper, it is proved that for $2<c<\frac{408}{197}$, the Diophantine equation $[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]=N$ is solvable in prime variables $p_1,\dots ,p_5$ for sufficiently large integer $N$.

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality

In this paper, we consider the Diophantine equation in the title, where $p,q$ are distinct odd prime numbers and $a,b$ are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of $p,q,a$ and $b$. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

We present an elementary construction of an explicit two-parametric family of elliptic curves over $\mathbb{Q}$ of the form $y^2=x^3+k$ such that the rank of each member of the family is at least seven. The main new ingredient is Ramanujan’s identity for sums of rational cubes.

We give an algorithm that produces all solutions of the equation ${\sum }_{i=1}^{n}1/x_i=1$ in integers of the form $2^a{k}^b$, where k is a fixed positive integer that is not a power of 2, a is an element of $\{0,1,2\}$ that can vary from term to term, and b is a nonnegative integer that can vary from term to term. We also completely characterize the pairs $(k,n)$ for which this equation has a nontrivial solution in integers of this form.

Unlike the problem of finding Eulerian cycles, the problem of deciding whether a given graph has a Hamiltonian cycle is NP-complete, even when restricting to planar graphs. There are however some criteria that can be used in special cases. One of them can be derived by Grinberg’s theorem. In this note, we give some more (slightly more general) criteria derived from Grinberg’s theorem to show that a graph does not have a Hamiltonian cycle.

In this paper, we study a special holomorphic family (M, π, R) of closed Riemann surfaces of genus two over a fourth punctured torus R, which is a kind of a Kodaira surface and is constructed by Riera. We give two explicit defining equations for (M, π, R) by using elliptic functions, and determine all the holomorphic sections of (M, π, R). Proofs will appear elsewhere.