Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations

Terai, Nobuhiro

doi:10.4064/aa-90-1-17-35

Cited by 37 publications

(24 citation statements)

References 18 publications

(1 reference statement)

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“…(α) (Terai [25]) b ≡ 3 (mod 8), b ≥ 30a and a d = −1, where d > 1 is a divisor of b and a d denotes the Jacobi symbol, (β) (Cao [4]) c is a prime power, (γ) (Cao-Dong [5]) b ≥ 25.1a, (δ) (Le [14]) c > 3 · 10 27 and r > 7200. Further partial confirmations of the conjecture are referred to in the papers cited above.…”

Section: The Problemmentioning

confidence: 99%

On a conjecture on exponential Diophantine equations

Cipu

Mignotte

2009

Acta Arith.

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We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution

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Section: The Problemmentioning

confidence: 99%

On a conjecture on exponential Diophantine equations

Cipu

Mignotte

2009

Acta Arith.

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“…As an analogue of Jeśmanowicz' conjecture, the author proposed that if , , , , , are fixed positive integers satisfying + = with , , , , , ≥ 2 and gcd( , ) = 1, then (1.1) has only the positive integer solution ( , , ) = ( , , ) except for a handful of triples ( , , ) (cf. [6,12,13,15,21,24]). This conjecture has been proved to be true in many special cases.…”

Section: Introductionmentioning

confidence: 99%

On the exponential Diophantine equation (4m²+1)ˣ + (21m²-1)ʸ = (5m)ᶻ

Terai¹

2020

AMI

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“…The theme of this note is to study a problem proposed by Terai [18], which origins at his paper published in 1994 (cf. [15]).…”

Section: Introductionmentioning

confidence: 99%

On Terai’s exponential equation with two finite integer parameters

Miyazaki¹

2019

NNTDM

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Let r be an integer with r > 1, and m be an even positive integer. Define integers A and B by the equation A + B √ −1 = (m + √ −1) r. It is proven by F. Luca in 2012 that the equation |A| x + |B| y = (m 2 + 1) z does not hold for any triple (x, y, z) of positive integers not equal to (2, 2, r), whenever r or m exceeds some effectively computable absolute constant. In our previous work, we estimated this constant explicitly. Here that estimate is substantially improved.

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Applications of a lower bound for linear forms in two logarithms to exponential Diophantine equations

Cited by 37 publications

References 18 publications

On a conjecture on exponential Diophantine equations

On a conjecture on exponential Diophantine equations

On the exponential Diophantine equation (4m²+1)ˣ + (21m²-1)ʸ = (5m)ᶻ

On Terai’s exponential equation with two finite integer parameters

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