Positive Integer Solutions Research Articles

All solutions of the Diophantine equation 1/x+1/y+1/z=u/(u+2)

In the history of mathematics, many mathematical researchers have investigated the Diophantine equation in the form , where and are positive integers. Without loss of generality, we may assume that . This Diophantine equation, also known as the Egyptian fraction equation of length 3, is to write the fraction as a sum of three fractions with the numerator being one and the denominators being different positive integers. Examples of research such as, in 2021, Sandor and Atanassov studied and found that the Diophantine equation has forty-four positive integer solutions. In this paper, we will study and find the complete positive integer solutions of the Diophantine equation , by using elementary methods of number theory and computer calculations. In the process, we can see that . Then, we will consider separately the value of a positive integer in nine cases. The first case is impossible. For the second and third cases, we will separate to consider the value of . For the remaining cases, we will separate to consider the value of . The research results showed that all positive integer solutions of the Diophantine equation are eighty-seven positive integer solutions. Moreover, from the steps to find the above positive integer solutions, we expect that it can be used to find the complete positive integer solutions of the Diophantine equation , where is a positive integer with .

Exposure Of Positive Integer Solutions to an Equation Comprising Kaprekar Numbers

Exposure Of Positive Integer Solutions to an Equation Comprising Kaprekar Numbers

Number of solutions to a special type of unit equations in two unknowns

abstract: For any fixed relatively prime positive integers $a$, $b$ and $c$ with $\min\{a,b,c\}>1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x$, $y$ and $z$, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.~A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), no.~2, 897--922] which asserts that Pillai's type equation $a^x-b^y=c$ has at most two solutions in positive integers $x$ and $y$ for any fixed positive integers $a$, $b$ and $c$ with $\min\{a,b\}>1$.

On the Diophantine equation σ2(X¯n) = σn(X¯n)

In this paper, we investigate the set [Formula: see text] of positive integer solutions of the title Diophantine equation. In particular, for a given [Formula: see text] we prove boundedness of the number of solutions, give precise upper bound on the common value of [Formula: see text] and [Formula: see text] together with the biggest value of the variable [Formula: see text] appearing in the solution. Moreover, we enumerate all solutions for [Formula: see text] and discuss the set of values of [Formula: see text] over elements of [Formula: see text].

On a conjecture concerning the exponential Diophantine equation $ (an^{2}+1)^{x}+(bn^{2}-1)^{y} = (cn)^{z} $

Let $ a $, $ b $, $ c $, and $ n $ be positive integers such that $ a+b = c^{2} $, $ 2\nmid c $ and $ n > 1 $. In this paper, we prove that if $ \gcd(c, n) = 1 $ and $ n\geq 117.14c $, then the equation $ (an^{2}+1)^{x}+(bn^{2}-1)^{y} = (cn)^{z} $ has only the positive integer solution $ (x, y, z) = (1, 1, 2) $ under the assumption $ \gcd(an^{2}+1, bn^{2}-1) = 1 $. Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer $ n $, if $ (a, b, c) = (12, 13, 5) $, $ (18, 7, 5) $, or $ (44, 5, 7) $, then this equation has only the solution $ (x, y, z) = (1, 1, 2) $. This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently.

Generalization of Markov Diophantine Equation via Generalized Cluster Algebra

In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_3xy+k_1yz+k_2zx=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+2xy^2+ky^2z^2+2xz^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.

Narayana sequence and the Brocard–Ramanujan equation

Let $\left\lbrace a_{n}\right\rbrace_{n\geq 0}$ be the Narayana sequence defined by the recurrence $a_{n}=a_{n-1}+a_{n-3}$ for all $n\geq 3$ with intital values $a_{0}=0$ and $a_{1}=a_{2}=1$. In this paper, we fully characterize the $3$-adic valuation of $a_{n}+1$ and $a_{n}-1$ and then we find all positive integer solutions $(u,m)$ to the Brocard--Ramanujan equation $m!+1=u^2$ where $u$ is a Narayana number.

A note on Dujella's unicity conjecture

Using properties of binary quadratic Diophantine equations, we prove that if \(r=p^{m} q^{n}\), where \(p, q\) are distinct odd primes and \(m, n\) are positive integers, then the equation \(x^{2}-\left(r^{2}+1\right) y^{2}=r^{2}\) has at most one positive integer solution \((x, y)\) with \(y \lt r-1\).

On Y-coordinates of Pell equations which are Fibonacci numbers

Let d ge 2 be an integer which is not a square. We show that if (F_n)_{nge 0} is the Fibonacci sequence and (X_m, Y_m)_{mge 1} is the mth solution of the Pell equation X^2 -dY^2 = pm 1, then the equation Y_m = F_n has at most two positive integer solutions (m, n) except for d=2 when it has three solutions (m,n)=(1,2),(2,3),(3,5).

Nonexistence of Positive Integer Solutions of the Diophantine Equation p^x + (p + 2q)^ y = z^2 , where p, q and p + 2q are Prime Numbers

The Diophantine equation p^x + (p + 2q)^y = z^2 , where p, q and p + 2q are prime numbers, is studied widely. Many authors give q as an explicit prime number and investigate the positive integer solutions and some conditions for non-existence of positive integer solutions. In this work, we gather some conditions for odd prime numbers p and q for showing that the Diophantine equation p^x + (p + 2q)^y = z^2 has no positive integer solution. Moreover, many examples of Diophantine equations with no positive integer solution are illustrated.

Mersenne version of Brocard-Ramanujan equation

In this study, we deal with a special form of the Brocard-Ramanujan equation, which is one of the interesting and still open problems of Diophantine analysis. We search for the positive integer solutions of the Brocard-Ramanujan equation for the case where the right-hand side is Mersenne numbers. By using the definition of Mersenne numbers, appropriate inequalities for the parameters of the equation, and the prime factorization of $n!$ we show that there is no positive integer solution to this equation. Thus, we obtain this interesting result demonstrating that the square of any Mersenne number can not be expressed as $n!+1$.

Degeneration of N-solitons and interaction of higher-order solitons for the (2+1)-dimensional generalized Hirota-Satsuma-Itoequation

In this paper, the evolutionary behavior of N-solitons for a (2+1)-dimensional generalized Hirota-Satsuma-Ito equation is studied by using the Hirota bilinear method and the long wave limit method. Based on the N-soliton solution, we first study the evolution from N-soliton to T-order (T = 1, 2, 3) breather wave solutions via the paired-complexification of parameters, and then we get the N-order rational solutions, M-order (M = 1, 2, 3) lump solutions, and the hybrid behavior between a variety of different types of solitons combined with the parameter limit technique and the paired-complexification of parameters. Besides, the interaction phenomenon between 1-order lump solution and N-soliton (N takes any positive integer) solution is investigated, and we give a computational proof process and an example. Meanwhile, we also provide a large number of three-dimensional and two-dimensional figures in order to better show the degeneration of the N-soliton and the interaction behavior between different N-solitons.

An elementary approach to the generalized Ramanujan–Nagell equation

Let k be a fixed positive integer with $$k>1$$ . In this paper, using various elementary methods in number theory, we give criteria under which the equation $$x^2+(2k-1)^y=k^z$$ has no positive integer solutions (x, y, z) with $$y\in \{3,5\}$$ .

Complete solutions of the simultaneous Pell's equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $

<abstract><p>In this paper, we consider the simultaneous Pell equations $ (a^2+2)x^2-y^2 = 2 $ and $ x^2-bz^2 = 1 $ where $ a $ is a positive integer and $ b > 1 $ is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation $ (a^2+2)x^2-y^2 = 2 $ are $ x = x_m $ and $ y = y_m $ with $ m\geq 1 $ odd, then the only solution of the system is given by $ m = 3 $ or $ m = 5 $ or $ m = 7 $ or $ m = 9 $.</p></abstract>

Jordan blocks and the Bethe Ansatz II: The eclectic spin chain beyond K = 1

We continue the classification of the Jordan chains of the eclectic three state spin chain that we started in our previous article. Following the same steps, we construct the generalised eigenvectors of this spin chain by computing the strongly twisted limit of linear combinations of eigenvectors of a twisted XXX SU(3) spin chain. We show that this classification problem can be mapped to the computation of the number of positive integer solutions of a system of linear equations.

ON A CONJECTURE CONCERNING THE NUMBER OF SOLUTIONS TO

AbstractLet a, b, c be fixed coprime positive integers with $\min \{ a,b,c \}>1$ . Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$ . We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples, then $c>10^{18}$ , and there are exactly two solutions $(x_1, y_1, z_1)$ , $(x_2, y_2, z_2)$ with $2 \mid x_1$ , $2 \mid y_1$ , $z_1=1$ , $2 \nmid y_2$ , $z_2>1$ , and, taking $a<b$ , we must have $a=2$ , $b \equiv 1 \bmod 12$ , $c \equiv 5\, \mod 12$ , with $(a,b,c)$ satisfying further strong restrictions. These results support a conjecture put forward by Scott and Styer [‘Number of solutions to $a^x + b^y = c^z$ ’, Publ. Math. Debrecen88 (2016), 131–138].

On the Exponential Diophantine Equation $(6m^{2}+1)^{x}+(3m^{2}-1)^{y}=(3m)^{z}$

Let $m$ be a positive integer. In this paper, we consider the exponential Diophantine equation $(6m^{2}+1)^{x}+(3m^{2}-1)^{y}=(3m)^{z}$ and we show that it has only unique positive integer solution $(x,y,z)=(1,1,2)$ for all $ m>1. $ The proof depends on some results on Diophantine equations and the famous primitive divisor theorem.

Sums of Fibonacci numbers that are perfect powers

Let us denote by Fn the n-th Fibonacci number. In this paper we show that for a fixed integer y there exists at most one integer exponent a > 0 such that the Diophantine equation Fn + Fm = ya has a solution (n, m, a) in positive integers satisfying n > m > 0, unless y = 2, 3, 4, 6 or 10.

The equation $x/y+y/z+z/x=4$ revisited

We give an elementary argument to the fact that the equation x/y+y/z+z/x=4 has no positive integer solutions.

A NOTE ON THE NUMBER OF SOLUTIONS OF TERNARY PURELY EXPONENTIAL DIOPHANTINE EQUATIONS

AbstractLet a, b, c be fixed coprime positive integers with $\min \{a,b,c\}>1$ . We discuss the conjecture that the equation $a^{x}+b^{y}=c^{z}$ has at most one positive integer solution $(x,y,z)$ with $\min \{x,y,z\}>1$ , which is far from solved. For any odd positive integer r with $r>1$ , let $f(r)=(-1)^{(r-1)/2}$ and $2^{g(r)}\,\|\, r-(-1)^{(r-1)/2}$ . We prove that if one of the following conditions is satisfied, then the conjecture is true: (i) $c=2$ ; (ii) a, b and c are distinct primes; (iii) $a=2$ and either $f(b)\ne f(c)$ , or $f(b)=f(c)$ and $g(b)\ne g(c)$ .