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Maohua Le

Publications and source records attributed to Maohua Le.

16 recordsLinked to original sources
arXiv2025

Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture

We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation \[ n^x=(n+1)^y+(n+2)^z+(n+3)^w \] in positive integers $n,x,y,z$ and $w$ are given by $(n,x,y,z,w)=(2,5,1,1,2)$, $(3,3,2,1,1)$. Our proof of this result for each $n \ge 4$ provides an explicit modulus $M$ such that the corresponding equation has no solution already modulo $M$. This contributes to a classical problem posed by T. Skolem in 1930's on a local-global principle on purely exponential Diophantine equations.

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arXiv2025

General sharp bounds for the number of solutions to purely exponential equations with three terms

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. In this paper, we prove that for any fixed $c$ there is at most one solution to the equation, except for only finitely many pairs of $a$ and $b.$ This is regarded as a 3-variable generalization of the result of Miyazaki and Pink [T. Miyazaki and I. Pink, Number of solutions to a special type of unit equations in two unknowns, III, arXiv:2403.20037 (accepted for publication in Math. Proc. Cambridge Philos. Soc.)] which asserts that for any fixed positive integer $a$ there are only finitely many pairs of coprime positive integers $b$ and $c$ with $b>1$ such that the Pillai's type equation $a^x-b^y=c$ has more than one solution in positive integers $x$ and $y$. The proof of our result is based on a certain $p$-adic idea of Miyazaki and Pink and relies on many deep theorems on the theory of Diophantine approximation, and it also includes the complete description of solutions to some interesting system of simultaneous polynomial-exponential equations. We also discuss how effectively exceptional pairs of $a$ and $b$ on our result for each $c$ can be determined.

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arXiv2023

A binary quadratic approach to $X^2+(2k-1)^Y=k^Z$

A conjecture of N. Terai states that for any integer $k>1$, the equation $x^2+(2k-1)^y =k^z$ has only one solution, namely, $(x, y, z) = (k-1, 1, 2).$ Using the structure of class groups of binary quadratic forms, we prove the conjecture when $4\Vert k$, with $2k-1$ a prime power and $4\le k\le 1000$.

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arXiv2023

On the Ternary Purely Exponential Diophantine Equation $(ak)^x+(bk)^y=((a+b)k)^z$ with Prime Powers $a$ and $b$

Let $k$ be a positive integer, and let $a,b$ be coprime positive integers with $\min\{a,b\}>1$. In this paper, using a combination of some elementary number theory techniques with classical results on the Nagell-Ljunggren equation, the Catalan equation and some new properties of the Lucas sequence (\seqnum{A000204} in OEIS), we prove that if $k>1$ and $a,b$ are both prime powers with $\min\{a,b\}>2$, then the equation $(ak)^x+(bk)^y=((a+b)k)^z$ has only one positive integer solution $(x,y,z)=(1,1,1)$. The above result partially proves that Conjecture 1 presented in (Acta Arith. 2018, 184 (1): 37-49) is true.

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arXiv2023

On a conjecture concerning the number of solutions to $a^x+b^y=c^z$, II

Let $a$, $b$, $c$ be distinct primes with $a 1$ and $(a,b,c)$ is not one of the six known such triples then $(a,b,c)$ must be one of three cases. In the present paper, we eliminate two of these cases (using the special properties of certain continued fractions for one of these cases, and using a result of Dirichlet on quartic residues for the other). Then we show that the single remaining case requires severe restrictions, including the following: $a=2$, $b \equiv 1 \bmod 48$, $c \equiv 17 \mod 48$, $b > 10^9$, $c > 10^{18}$; at least one of the multiplicative orders $u_c(b)$ or $u_b(c)$ must be odd (where $u_p(n)$ is the least integer $t$ such that $n^t \equiv 1 \bmod p$); 2 must be an octic residue modulo $c$ except for one specific case; $2 \mid v_2(b-1) \le v_2(c-1)$ (where $v_2(n)$ satisfies $2^{v_2(n)} \parallel n$); there must be exactly two solutions $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ with $1 = z_1 < z_2$ and either $x_1 \ge 28$ or $x_2 \ge 88$. These results support a conjecture put forward in \cite{ScSt6} and improve results in \cite{LeSt}.

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arXiv2022

On a conjecture concerning the number of solutions to $a^x+b^y=c^z$

Let $a$, $b$, $c$ 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, taking $a 10^{14}$. These results support a conjecture of the last two authors.

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arXiv2022

A modular approach to the generalized Ramanujan-Nagell equation

Let $k$ be a positive integer. In this paper, using the modular approach, we prove that if $k\equiv 0 \pmod{4}$, $30< k<724$ and $2k-1$ is an odd prime power, then under the GRH, the equation $x^2+(2k-1)^y=k^z$ has only one positive integer solution $(x,y,z)=(k-1,1,2)$. The above results solve some difficult cases of Terai's conecture concerning this equation.

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arXiv2021

On the power values of the sum of three squares in arithmetic progression

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ (*) when $n$ is an odd prime and $d=p^r$, $p>3$ a prime. So this improves the results on the papers of A. Koutsianas and V. Patel \cite{KP} and A. Koutsianas \cite{Kou}. Secondly, under the assumption of our first result, we prove that (*) has at most one solution $(x,y)$. Next, for a general $d$, we prove the following two results: (i) if every odd prime divisor $q$ of $d$ satisfies $q\not\equiv \pm 1 \pmod{2n},$ then (*) has only the solution $(x,y,d,n)=(21,11,2,3)$. (ii) if $n>228000$ and $d>8\sqrt{2}$, then all solutions $(x,y)$ of (*) satisfy $y^n<2^{3/2}d^3$.

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arXiv2020

A note on the ternary purely exponential Diophantine equation $A^x+B^y=C^z$ with $A+B=C^2$

Let $\ell, m, r$ be fixed positive integers such that $2\nmid \ell$, $3\nmid \ell m$, $\ell>r$ and $3\mid r$. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if $\min\{r\ell m^2-1,(\ell-r)\ell m^2+1\}>30$, then the equation $(r\ell m^2-1)^x+((\ell -r)\ell m^2+1)^y=(\ell m)^z$ has only the positive integer solution $(x,y,z)=(1,1,2)$.

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arXiv2018

An application of the BHV theorem to a new conjecture on exponential diophantine equations

Let $A$, $B$ be fixed positive integers such that $\min\{A,B\} > 1$, $\gcd(A,B) = 1$ and $AB \equiv 0 \bmod 2$, and let $n$ be a positive integer with $n>1$. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier \cite{BHV}, we prove that if $A > 8 B^3$, then the equation $(*) \quad (A^2 n)^x + (B^2 n)^y = ((A^2 + B^2)n)^z$ has no positive integer solutions $(x,y,z)$ with $x > z > y$. Combining the above conclusion with some existing results, we can deduce that if $A >8 B^3$ and $B \equiv 2 \bmod 4$, then (*) has only the positive integer solution $(x,y,z) = (1,1,1)$.

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arXiv2018

An application of Baker's method to the Jeśmanowicz' conjecture on primitive Pythagorean triples

Let $m$, $n$ be positive integers such that $m>n$, $\gcd(m,n)=1$ and $m \not\equiv n \bmod 2$. In 1956, L. Jeśmanowicz \cite{Jes} conjectured that the equation $(m^2 - n^2)^x + (2mn)^y = (m^2+n^2)^z$ has only the positive integer solution $(x,y,z) = (2,2,2)$. This problem is not yet solved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent \cite{Lau} with some elementary methods, we prove that if $mn \equiv 2 \bmod 4$ and $m > 30.8 n$, then Jeśmanowicz' conjecture is true.

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