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Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers

- Y. Bugeaud, M. Mignotte, S. Siksek
- Mathematics
- 2 March 2004

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based… Expand

Infinite Descent on Elliptic Curves

- S. Siksek
- Mathematics, Computer Science
- 1 December 1995

We present an algorithm for computing an upper bound for the difference of the logarithmic height and the canonical height on elliptic curves. Moreover a new method for performing the infinite… Expand

Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation

- Y. Bugeaud, M. Mignotte, S. Siksek
- MathematicsCompositio Mathematica
- 12 May 2004

This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based… Expand

Elliptic curves over real quadratic fields are modular

- Nuno Freitas, Bao V. Le Hung, S. Siksek
- Mathematics
- 26 October 2013

We prove that all elliptic curves defined over real quadratic fields are modular.

The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields

- Nuno Freitas, S. Siksek
- MathematicsCompositio Mathematica
- 11 July 2013

Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions… Expand

Height difference bounds for elliptic curves over number fields

- J. Cremona, M. Prickett, S. Siksek
- Mathematics
- 2006

Fermat's Last Theorem over some small real quadratic fields

- Nuno Freitas, S. Siksek
- Mathematics
- 16 July 2014

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the… Expand

Perfect powers that are sums of consecutive cubes

- M. Bennett, V. Patel, S. Siksek
- Mathematics
- 29 March 2016

Euler noted the relation 63 = 33 + 43 + 53 and asked for other
instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas,… Expand

On the asymptotic Fermat’s last theorem over number fields

- Mehmet Haluk Sengun, S. Siksek
- MathematicsCommentarii Mathematici Helvetici
- 14 September 2016

Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a… Expand

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