A NOTE ON THE DIOPHANTINE EQUATION
2018 ◽
Vol 98
(2)
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pp. 188-195
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Keyword(s):
Let $c\geq 2$ be a positive integer. Terai [‘A note on the Diophantine equation $x^{2}+q^{m}=c^{n}$’, Bull. Aust. Math. Soc.90 (2014), 20–27] conjectured that the exponential Diophantine equation $x^{2}+(2c-1)^{m}=c^{n}$ has only the positive integer solution $(x,m,n)=(c-1,1,2)$. He proved his conjecture under various conditions on $c$ and $2c-1$. In this paper, we prove Terai’s conjecture under a wider range of conditions on $c$ and $2c-1$. In particular, we show that the conjecture is true if $c\equiv 3\hspace{0.6em}({\rm mod}\hspace{0.2em}4)$ and $3\leq c\leq 499$.
2014 ◽
Vol 90
(1)
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pp. 9-19
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2015 ◽
Vol 11
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pp. 1107-1114
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2020 ◽
Vol 57
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pp. 200-206
2020 ◽
Vol 16
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pp. 1701-1708
2012 ◽
Vol 241-244
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pp. 2650-2653
2018 ◽
Vol 14
(05)
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pp. 1223-1228
2021 ◽
Vol 29
(3)
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pp. 23-32
2017 ◽
Vol 55
(1)
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pp. 115-118