Two Equations Involving the Smarandache LCM Dual Function

Pillai's conjecture revisited

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]

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction $S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}$. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.

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.

<p>Abstract For any positive integer n, the famous Smarandache function S(n) is defined as the smallest positive integer m such that nIm!.</p>

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.

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 $2 \\leq c \\leq 30$.

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.

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 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 $(x,y,z) = (2,2l-1,1)$.

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

In this paper we prove that the equation (2 n − 1)(6 n − 1) = x 2 has no solutions in positive integers n and x. Furthermore, the equation (a n − 1) (a kn − 1) = x 2 in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).