Recipes to Fermat-type equations of the form x^r + y^r = Cz^p

Details Recipes to Fermat-type equations of the form x^r + y^r = Cz^p Recipes to Fermat-type equations of the form x^r + y^r = Cz^p

2012-03-15

arxiv.org/abs/1203.3371v1

Mathematics

Number Theory

25 pages

Scientific paper

In this paper we discuss a general approach to Diophantine equations of the form x^r + y^r = Cz^p via Hilbert modular forms over some totally real subfields of the cyclotomic field Q(\zeta_r). In particular, we will prove for r=7 the non-existence of primitive solutions (a,b,c) such that 7 do not divide c and give explicit Frey-curves for r=11,13,17,19. Furthermore, for primes r=4m+1 we will give an extra method to construct two more Frey-curves.

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