A note on Dujella's unicity conjecture

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]

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.

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).

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 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.

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.

In this paper, we study the Diophantine equation a^x+(a+5b)^y=z^2 when a\equiv 1\left(\bmod 5\right) and b is a positive integer. We establish that the equation has no solutions in positive integers x,y and z. We start with the Diophantine equation p^x+(p+5a)^y=z^2 where p and p+5a are both primes and p\equiv 1\left(\bmod 5\right) and a is a positive integer.

Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2x + by = cz is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2r − 1, r + 2,2, 2), where r is a positive integer with r ≥ 2.

Let $ a $, $ b $, $ c $, and $ n $ be positive integers such that $ a+b = c^{2} $, $ 2\nmid c $ and $ n &amp;gt; 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.

Let $a$, $b$, $c$ be fixed coprime positive integers. In this paper we prove that if $b \equiv 3 \pmod{4}$, $a \equiv -1 \pmod{b^{2l}}$, $a^2 + b^{2l-1} = c$ and $c$ is odd, where $l$ is a positive integer, then the equation $a^x + b^y = c^z$ has only the positive integer solution $(x,y,z) = (2,2l-1,1)$.

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.

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.

We apply the hypergeometric method of Thue and Siegel to prove that if a and b are positive integers, then the inequality 0 &lt; |ax – by | &lt; ¼ max ﹛ax/2, by/2﹜ has at most a single solution in positive integers x and y. This essentially sharpens a classic result of LeVeque.

On the number of solutions of binomial Thue inequalities