Asymptotic Fermat for signatures (r, r, p) using the modular approach

Abstract

Abstract Let K be a totally real field, and $$r\ge 5$$ r ≥ 5 a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions $$(x,y,z) \in \mathcal {O}_K^3$$ ( x , y , z ) ∈ O K 3 of the signature ( r , r , p ) equation $$x^r+y^r=z^p$$ x r + y r = z p (where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of $$K(\zeta _r)$$ K ( ζ r ) . When $$K=\mathbb {Q}$$ K = Q we get that for most of the primes $$r<150$$ r < 150 with $$r \not \equiv 1 \mod 8$$ r ≢ 1 mod 8 there are no non-trivial, primitive integer solutions ( x , y , z ) with 2| z for signatures ( r , r , p ) when p is sufficiently large. Similar results hold for quadratic fields, for example when $$K=\mathbb {Q}(\sqrt{2})$$ K = Q ( 2 ) there are no non-trivial, primitive solutions $$(x,y,z)\in \mathcal {O}_K^3$$ ( x , y , z ) ∈ O K 3 with $$\sqrt{2}|z$$ 2 | z for signatures (5, 5, p ), (11, 11, p ), (13, 13, p ) and sufficiently large p .