Catalan’s Conjecture: Another old Diophantine problem solved

<p>Let f(n)=1 if n=1, 2^(2^(n-2)) if n \\in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \\in {6,7,8,...}. We conjecture that if a system T \\subseteq {x_i+1=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \\leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite. We show that the conjecture and Matiyasevich's conjecture on finite-fold Diophantine representations are jointly inconsistent.</p>

Suppose that $c$, $m$, and $a$ are positive integers with $a \equiv 11,\,13 \pmod{24}$. In this work, we prove that when $2c+1=a^{2}$, the Diophantine equation in the title has only solution $(x, y, z)=(1,1,2)$ where $m \equiv \pm 1 \pmod{a}$ and $m>a^2$ in positive integers. The main tools of the proofs are elementary methods and Baker's theory.

Linear Diophantine equation are introduced to determine and search for integral solutions according to the associated variables. In this paper, the mathematical techniques are approached to solve Linear Diophantine equation with two, three unknowns and system of linear Diophantine equation by Euclidean algorithmic, congruence modulo n and unimodular row reduction. The modulo arithmetic operation is also applied for linear Diophantine equation in three variables. The Euclidean algorithmic is an efficient method to compute the greatest common divisor (g.c.d) of two integers. The extended Euclidean algorithmic is used also to compute the greatest common divisor (g.c.d) of two positive integers and write this greatest common divisor as an integer linear combination of two positive integers. Moreover, the mathematical methods of Diophantine equations in two and three variables are presented to practical applications in reality life such as buying and selling, traffic flux and the number of atoms of a chemical substance to investigate positive integral solutions.

In this paper, we study a balancing problem of ordinary binomial coefficients. The diop-hantine equation [Formula: see text] in positive integers x and y ≥ x + 2 will be investigated, where k and ℓ are given positive integers. In case of k < ℓ the basic tool is the Runge method. If k ≥ ℓ, then the main results state effective and non-effective finiteness theorems on the subject matter. Further, computational results are also reported including the complete solutions when 1 ≤ k, ℓ ≤ 3, and a computer search otherwise.

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

Let $q$ be an odd prime. Let $c>1$ and $t$ be positive integers such that $q^{t}+1=2c^{2}$. Using elementary method and a result due to Ljunggren concerning the Diophantine equation $\\frac{x^{n}-1}{x-1}= y^{2}$, we show that the Diophantine equation $x^{2}+q^{m}=c^{2n}$ has the only positive integer solution $(x, m, n)=(c^{2}-1, t, 2)$. As applications of this result some new results on the Diophantine equation $x^{2}+q^{m} = c^{n}$ and the Diophantine equation $x^{2}+(2c-1)^{m} = c^{n}$ are obtained. In particular, we prove that Terai’s conjecture is true for $c=12,24$. Combining this result with Terai’s results we conclude that Terai’s conjecture is true for $2 \\leq c \\leq 30$.

This paper investigates the solutions (if any) of the Diophantine equation 3x + 35y = Z2, where , x, y, and z are whole numbers. Diophantine equations are drawing the attention of researchers in diversified fields over the years. These are equations that have more unknowns than a number of equations. Diophantine equations are found in cryptography, chemistry, trigonometry, astronomy, and abstract algebra. The absence of any generalized method by which each Diophantine equation can be solved is a challenge for researchers. In the present communication, it is found with the help of congruence theory and Catalan’s conjecture that the Diophantine equation 3x + 35y = Z2 has only two solutions of (x, y, z) as (1, 0, 2) and (0, 1, 6) in non-negative integers.

The classification of tensor categories of two-colored noncrossing partitions

We prove that if a complex valued completely multiplicative function F and a positive integer l≦5 satisfy the condition F(N) = Ul, where Ul is the set of all l-th roots of unity, then {F(n+1) F(n) ∣ ne N} = Ul.

Let K be an absolutely abelian number field, and t be a positive integer relatively prime to $[K: \mathbb {Q}]$ . We provide a method to use the t -rank of the class group of K to obtain a lower bound on the t -rank of class groups of any cyclotomic fields (or real cyclotomic fields) that contain this K . As an application, for any odd integer $t> 1$ , we provide a family of cyclotomic fields whose class groups have t -rank at least $2$ . Furthermore, we prove that the class number of $\mathbb {Q}(\zeta _{p^e} + \zeta _{p^e}^{-1})$ cannot be equal to $2$ , where p is a prime, $e \geq 1$ is an integer, and $\zeta _{p^e}$ denotes a primitive $p^e$ -th root of unity. We also provide an upper bound for the class number of a real cyclotomic field.

Since Diophantine equations play an important role in solving important real-world problems such as business investment problems, network flow problems, pole placement problems, and data privacy problems, researchers are increasingly interested in developing new techniques for analyzing the nature and solutions of the various Dio phantine equations. In this study we investigate the Diophantine problem 22x+40y = z2, where x, y, z are non-negative integers, and discover that it does not have a non-negative integer solution.

Jeśmanowicz [7] conjectured that the exponential Diophantine equation \({{(m^2-n^2)}^x + {(2mn)}^y = {(m^2 + n^2)}^z}\) has only one positive integer solution (x, y, z) = (2, 2, 2), where m and n are positive integers with m > n, gcd(m, n) = 1 and m ≢ n mod 2. In [17], the second author showed that if n = 2, then Jeśmanowicz’ conjecture is true without any assumption on m. Let n be a positive integer satisfying \({n \equiv 2\mod 4}\) and m > 72n. We prove that if n satisfies at least one of three conditions \({(\mathrm{C}_1)\,n/2}\) is a power of an odd prime, \({(\mathrm{C}_2) n/2}\) has no prime factors congruent to 1 modulo 8, \({(\mathrm{C}_3) n/2}\) is a square, then the conjecture is true. As a corollary, we derive that if n/2 is odd with n < 100, then the conjecture is true without any assumption on m.

Max–min of polynomials and exponential diophantine equations

In this paper, we give a straightforward approach to obtaining the solution of the Diophantine equation frac{1}{w} + frac{1}{x} + frac{1}{y} + frac{1}{z} = frac{1}{2}. We also establish that the Diophantine equation frac{1}{w} + frac{1}{x} + frac{1}{y} + frac{1}{z} = frac{m}{n} for any two positive integers m and n has only a finite number of solutions in the positive integers w, x, y, and z.

Major progress in group theory occurred in the nineteenth century but the evolution of group theory began already in the latter part of 18th century. Some characteristic features of 19th century mathematics which had crucial impact on this evolution are concern for rigor and abstraction and the view that mathematics is a human activity without necessarily referring to physical situations. In 1770, Joseph Louis Lagrange (1736-1813) wrote his seminal memoir Reflections on the solution of algebraic equations. He considered ‘abstract’ questions like whether every equation has a root and, if so, how many were real/complex/positive/negative? The problem of algebraically solving 5th degree equations was a major preoccupation (right from the 17th century) to which Lagrange lent his major efforts in this paper. His beautiful idea (now going under the name of Lagrange’s resolvent) is to ‘reduce’ a general equation to auxiliary (resolvent) equations which have one degree less. Later, the theory of finite abelian groups evolved from Carl Friedrich Gauss’s famous “Disquisitiones Arithmeticae”. Gauss (1777-1855) established many of the important properties though he did not use the terminology of group theory. In his work, finite abelian groups4 Math Unlimitedappeared in different forms like the additive group Zn of integers modulo n, the multiplicative group Z∗n of integers modulo n relatively prime to n, the group of equivalence classes of binary quadratic forms, and the group of n-th roots of unity. In 1872, Felix Klein delivered a famous lecture A Comparative Review of Recent Researches in Geometry. The aim of his (so-called) Erlangen Program was the classification of geometry as the study of invariants under various groups of transformations. So, the groups appear here “geometrically” as groups of rigid motions, of similarities, or as the hyperbolic group etc. During the analysis of the connections between the different geometries, the focus was on the study of properties of figures invariant under transformations. Soon after, the focus shifted to a study of the transformations themselves. Thus the study of the geometric relations between figures got converted to the study of the associated transformations. In 1874, Sophus Lie introduced his “theory of continuous transformation groups” what we basically call Lie groups today. Poincaré and Klein began their work on the so-called automorphic functions and the groups associated with them around 1876. Automorphic functions are generalizations of the circular, hyperbolic, elliptic, and other functions of elementary analysis. They are functions of a complex variable z, analytic in some domain, and invariant under the group of transformations x′ = ax+b(a, b, c, d real or complex and ad − bc 0), or under some subgroup of this group. We end this introduction with the ancient problem of finding all those positive integers (called congruent numbers) which are areas of right-angled triangles with rational sides. To this date, the general result describing all of them precisely is not proved. However, the problem can be re-stated in terms of certain groups called elliptic curves and this helps in getting a hold on the problem and in obtaining several partial results. Indeed, if n is a positive integer, then the rational numbers x, y satisfying y2 = x3 − n2x form a group law which can be described geometrically. Then, n is a congruent number if and only if, this group is infinite!