Rational points on Erdős–Selfridge superelliptic curves
2016 ◽
Vol 152
(11)
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pp. 2249-2254
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Given $k\geqslant 2$, we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$) for which $$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$ In particular, if we assume that $\ell$ is prime, then all such triples $(x,y,\ell )$ satisfy either $y=0$ or $\ell <\exp (3^{k})$.
1980 ◽
Vol 21
(3)
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pp. 463-470
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2011 ◽
Vol 85
(1)
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pp. 105-113
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2018 ◽
Vol 482
(4)
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pp. 385-388
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Keyword(s):
1962 ◽
Vol 14
◽
pp. 565-567
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Keyword(s):
1989 ◽
Vol 41
(2)
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pp. 285-320
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2015 ◽
Vol 145
(6)
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pp. 1153-1182
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Keyword(s):
1994 ◽
Vol 116
(3)
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pp. 385-389
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Keyword(s):
1967 ◽
Vol 63
(1)
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pp. 83-85
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Keyword(s):