Elliptic curves over real quadratic fields are modular | SpringerLink Skip to main content
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Let K be a totally real Galois number field and let E be a set of elliptic curves over K. We give
sufficient conditions for the existence of a finite computable set of rational primes P such that …

Let $ K $ be a totally real number field with narrow class number one and $ O_K $ be its ring
of integers. We prove that there is a constant $ B_K $ depending only on $ K $ such that for …

For any fixed relatively prime positive integers $ a $, $ b $ and $ c $ with $\min\{a, b, c\}> 1$,
we prove that the equation $ a^ x+ b^ y= c^ z $ has at most two solutions in positive integers …

1. Introduction. Let a, b and c be positive integers satisfying a2+ b2= c2. Such a triple (a, b, c)
is called a Pythagorean triple. If gcd (a, b, c)= 1, this triple is called primitive. It is well-known …

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^ 3+ x^
3+(x+ 1)^ 3= z^ 2$ in integers $ x $, $ z $. The generalization $(x-1)^ k+ x^ k+ (x+ 1)^ k= z^ n …

The Generalized Fermat Equation | SpringerLink Skip to main content Advertisement
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We show that the equation a 2+ b 6= cn has no nontrivial positive integer solutions with (a,
b)= 1 via a combination of techniques based upon the modularity of Galois representations …

We describe a strategy to attack infinitely many Fermat-type equations of signature (r, r, p)(r,
r, p), where r ≥ 7 r≥ 7 is a fixed prime and pp is a prime allowed to vary. Indeed, to a …

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2+y 2m=zp
Page 1 Algebra & Number Theory msp Volume 10 2016 No. 6 Modular elliptic curves over real …