An upper bound for the number of solutions of ternary purely exponential Diophantine equations II

Hu, Yueyun; Mao-hua, LE

doi:10.5486/pmd.2019.8444

Cited by 9 publications

(11 citation statements)

References 5 publications

(6 reference statements)

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“…The proof of this proposition is achieved by the combination of Baker's method in both complex and p-adic cases together with improving the gap principle established by Hu and Le [HuLe,HuLe2,HuLe3] arising from the existence of three hypothetical solutions as well as a certain 2-adic argument relying upon the striking result of Scott and Styer [ScSt2]. From the viewpoint of the generalized Fermat equation (cf.…”

Section: Introductionmentioning

confidence: 99%

Number of solutions to a special type of unit equations in two variables II

Miyazaki¹,

Pink²

2022

Preprint

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This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers a, b and c 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. The fundamental result proves the conjecture under some congruence condition modulo c on a and b. As applications the conjecture is confirmed to be true if c takes some small values including the Fermat primes found so far, and this particularly provides an analytic proof of the celebrated theorem of Scott [R. Scott, On the equations p x − b y = c and a x + b y = c z , J. Number Theory 44(1993), 153-165] solving the conjecture for c = 2 in a purely algebraic manner. The method can be generalized for smaller modulus cases, and it turns out that the conjecture holds true for infinitely many specific values of c not being perfect powers. The main novelty is to apply a special type of the p-adic analogue to Baker's theory on linear forms in logarithms via a certain divisibility relation arising from the existence of two hypothetical solutions to the equation. The other tools include Baker's theory in the complex case and its non-Archimedean analogue for number fields together with various elementary arguments through rational and quadratic numbers, and extensive computation.

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Section: Introductionmentioning

confidence: 99%

Number of solutions to a special type of unit equations in two variables II

Miyazaki¹,

Pink²

2022

Preprint

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“…LEMMA 2.1. If c is an odd prime, then, for a given parity class, there is at most one solution (x, y, z) to (1.1), except for (a, b, c) = (3, 10, 13) or (10,3,13).…”

Section: Preliminariesmentioning

confidence: 99%

On a Conjecture Concerning the Number of Solutions To

Mao-hua¹,

Styer

2022

Bull. Aust. Math. Soc.

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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 $c>10^{18}$ , 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].

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“…By combining the Gel'fond-Baker method with an elementary approach, Hua and Le [15] proved that if max{a, b, c} > 5 × 10 27 , then the equation a x + b y = c z has at most three positive integer solutions (x, y, z). Moreover they showed that if max{a, b, c} ≥ 10 62 , then the equation (1.1) has at most two solutions in positive integers x, y and z [16]. It seems that the latter result holds with no restriction on values a,b and c. Scott and Styer made even a stronger conjecture, i.e.…”

Section: Introductionmentioning

confidence: 98%

On The Exceptional solutions of Jeśmanowicz' conjecture

Ghadermarzi¹

2020

Preprint

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Let (a, b, c) be a primitive Pythagorean triple. Set a = m 2 −n 2 , b = 2mn , and c = m 2 + n 2 with m and n positive coprime integers, m > n and m ≡ n (mod 2). A famous conjecture of Jeśmanowicz asserts that the only positive integer solution to the Diophantine equation a x + b y = c z is (x, y, z) = (2, 2, 2). In this note, we will prove that for any n > 0 there exists an explicit constant c(n) such that if m > c(n), the above equation has no exceptional solution when all x,y and z are even. Our result improves that of Fu and Yang [11]. As an application, we will show that if 4 || m and m > C(n), then Jeśmanowicz' conjecture holds.c r with some conditions on p,q, and r, then disapprove the possibility of the existence of any other solution.There are much work, results, and conjectures by several authors in this direction (e.g., [34], [5] , [6]...), which lead to the following conjecture [21]:Conjecture 1.1. For given coprime integers a, b, c > 1, the Diophantine equation (1.1) has at most one solution in integers x, y, z > 1.The most famous, old example is when a, b, and c are primitive Pythagorean triple, and hence p = q = r = 2 is a solution. Sierpiński [33] showed that the equation 3 x + 4 y = 5 z has a unique solution (2, 2, 2) in positive integers. In the same year, Jeśmanowicz [18] proved the same result for the Pythagorean triple (a, b, c) = (5, 12, 13), (7, 24, 25), (9, 40, 41) and (11; 60; 61). He further made the following conjecture: Conjecture 1.2. Let (a, b.c) be positive integers where a 2 +b 2 = c 2 and GCD(a, b) = 1. Then the Diophantine equation a x + b y = c z has only the positive integer solution (x, y, z) = (2, 2, 2).

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An upper bound for the number of solutions of ternary purely exponential Diophantine equations II

Cited by 9 publications

References 5 publications

Number of solutions to a special type of unit equations in two variables II

Number of solutions to a special type of unit equations in two variables II

On a Conjecture Concerning the Number of Solutions To

On The Exceptional solutions of Jeśmanowicz' conjecture

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