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Your search keyword '"Terai's conjecture"' showing total 2 results
Start Over Descriptor "Terai's conjecture" Topic 11d61
2 results on '"Terai's conjecture"'

1. On some ternary pure exponential diophantine equations with three consecutive positive integers bases.

Author

Fu, Ruiqin, He, Bo, Yang, Hai, and Zhu, Huilin

Subjects
*DIOPHANTINE equations, *INTEGERS, *LOGARITHMS, *MATHEMATICS
Abstract

By using the lower bound of linear forms in two logarithms of Laurent (Acta Arith. 133(4) (2008) 325–348), we give here a new solution that the ternary pure exponential diophantine equation (n + 1) x + (n + 2) y = n z has no positive integer solutions except for (n , x , y , z) = (3 , 1 , 1 , 2) . This proof is very different from Le (J. Yulin Teachers College28(3) (2007) 1–2), in which he used the classification method of solutions of exponential decomposition form equation. Furthermore, we solved completely another similar ternary pure exponential diophantine equation n x + (n + 2) y = (n + 1) z by using m-adic estimation of linear forms due to Bugeaud (Compos. Math. 132(2) (2002) 137–158). [ABSTRACT FROM AUTHOR]

Published
2019
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2. A note on the Diophantine equation $x^{2} + q^{m} = c^{2n}$

Author

Mou-Jie Deng

Subjects
Combinatorics, 11D61, integer solution, Conjecture, Integer, Diophantine set, Diophantine equations, General Mathematics, Diophantine equation, Terai’s conjecture, Prime (order theory), Mathematics
Abstract

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 $2 \leq c \leq 30$.

Published
2015
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