A note on the Diophantine equation $x^{2} + q^{m} = c^{2n}$

This paper proves that for all positive integers n, the equation $$ \frac{x}{y}+p\frac{y}{z}+\frac{z}{w}+p\frac{w}{x}=8np, $$ where p = 1 or p is a prime congruent to 1 (mod8), does not have solutions in positive integers.

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]

<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>

Pillai's conjecture revisited

Let $S_k(m):=\sum_{j=1}^{m-1}j^k$ denote a power sum. In 2011, Kellner proposed the conjecture that for $m>3$ the ratio $S_k(m+1)/S_k(m)$ is never an integer, or, equivalently, that for any positive integer $a$, the equation $aS_k(m)=m^k$ has no solutions in positive integers $k$ and $m$ with $m>3$. In this paper, we show that for many integers $a$ the equation $aT_k(m)=(2m+1)^k$, where $T_k(m):=\sum_{j=1}^m(2j-1)^k$, has no solutions in positive integers $k$ and $m$. This leads us to the conjecture that for $m>1$ the ratio $T_k(m+1)/T_k(m)$ is never an integer.

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.

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&amp;gt;1. $ The proof depends on some results on Diophantine equations and the famous primitive divisor theorem.

An algorithm is given for determining whether or not a finite system of conditions of the types a | B , a &gt; B a|B,a &gt; B , a is a square, possess a simultaneous solution in positive integers. Various generalizations are also obtained.

Let r be a positive integer with r > 1, and let m be a positive even integer. Further let a=|V(m, r)|, b=|U (m, r)| and c=m2+1, where V (m, r) + U (m, r)√−1=(m+√−1)r . In this paper, using the Gel′ fond-Baker method, we prove that if 2 | r and m>max(1015, 2r3), then the equation ax+by=cz has only the positive integer solution (x, y, z)=(2, 2, r).

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x 2 + (3n 2 + 1)y = (4n 2 + 1)z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n 3 + 3n, 1, 3).

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 \(y \lt r-1\).

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.

This study investigates the positive integer solutions of an equation involving the Smarandache function. The equation is given by tφ(n) = φ 2 (n) + S(n 16), where φ(n) represents the Euler function, φ e (n) represents the generalized Euler function with e as a positive integer, and S(n) represents the Smarandache function. The solutions of this equation are discussed, and it is proven that the equation only has positive integer solutions when t = 1, 2, 3, 6, 7, 9, 12, 17, 18, 19. Furthermore, all positive integer solutions of the equation are provided.

The Undecidability of Exponential Diophantine Equations

&lt;p&gt;The main purpose of this&nbsp;paper is using the elementary method to study the number of the solutions of two equations&nbsp;involving the Smarandache LCM dual function SL&curren;(n), and give their all positive integer&nbsp;solutions.&lt;/p&gt;