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Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
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
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
Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation
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
Elliptic curves over real quadratic fields are modular
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
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
Fermat's Last Theorem over some small real quadratic fields
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
Perfect powers that are sums of consecutive cubes
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,
On the asymptotic Fermat’s last theorem over number fields
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
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