On ternary Diophantine equations of signature(p,p,2)over number fields

Abstract

Let K be a totally real number field with narrow class number one and OK be its ring of integers. We prove that there is a constant BK depending only on K such that for any prime exponent p > BK the Fermat type equation x p + y p = z 2 with x, y, z ∈ OK does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

Asymptotic Fermat for signatures $(p,p,2)$ and $(p,p,3)$ over totally real fields

“…Also, recently in [30] and [31] T , urcas , studied Fermat equation over imaginary quadratic field Q( √ −d) with class number one. We now present a result by Işik, Kara and Ozman, proved in [16] which serves as the starting point of this paper. It gives a computable criteria of testing if the asymptotic Fermat Last Theorem holds for certain type of solutions of the equations with signatures (p, p, 2).…”

“…In [16], Işik, Kara and Ozman list all known cases where equation (1) has been solved over the rational integers in two tables (p.4). Table 1 contains all unconditional results for infinitely many primes.…”

“…This is a key result in the proof of Theorem 3, for which we have prepared the ingredients in the previous sections. We will follow the corresponding proofs in [12] and [16].…”

“…We will show that v P(j E ′ ) ≥ 0, contradicting Theorem 36 (iv) and hence we can conclude the proof. By using (16) we can rewrite the assumption |v P( α β )| ≤ 3v P(3) in terms of the valuation of µ, using that α β = µ 27 :…”

On ternary Diophantine equations of signature over number fields

“…In [18] we used the -unit equation method and this restricted us to the totally real number fields. However using Theorem 4.1 as we did for proving Theorem 1.…”