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

100%

Terai N.

Acta Arithmetica

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1999

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tom 90

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nr 1

17-35

2

The Diophantine equation $x^2 + q^m =p^n$

100%

Terai N.

Acta Arithmetica

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1993

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tom 63

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nr 4

351-358

3

Generators and integer points on the elliptic curve y² = x³ - nx

63%

Fujita Y., Terai N.

Acta Arithmetica

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2013

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tom 160

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nr 4

333-348

EN

Let E be an elliptic curve over the rationals ℚ given by y² = x³ - nx with a positive integer n. We consider first the case where n = N² for a square-free integer N. Then we show that if the Mordell-Weil group E(ℚ ) has rank one, there exist at most 17 integer points on E. Moreover, we show that for some parameterized N a certain point P can be in a system of generators for E(ℚ ), and we determine the integer points in the group generated by the point P and the torsion points. Secondly, we consider the case where n = s⁴ + t⁴ for distinct positive integers s and t. We then show that if n is fourth-power-free, the points P₁ = (-t²,s²t) and P₂ = (-s²,st²) can be in a system of generators for E(ℚ ). Furthermore, we prove that if n is square-free, then there exist at most nine integer points in the group Γ generated by the points P₁, P₂ and the torsion point (0,0). In particular, in case n = s⁴ + 1 the group Γ has exactly seven integer points.

Ograniczanie wyników

3 Acta Arithmetica

3 Terai N.

1 Fujita Y.

1 2013

1 1999

1 1993

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

The Diophantine equation $x^2 + q^m =p^n$

Generators and integer points on the elliptic curve y² = x³ - nx