Generalized Fermat equation over number fields

Abstract

Let K be a number field and A,B,C∈𝒪_K∖{0}. Put S_K={P:P|2} and S_K'={q:q|2ABC}. We establish two criteria for the non-existence of non-trivial solutions of Ax^p+By^p=Cz^p over K for sufficiently large prime exponents p. Both criteria use a valuation bound on the S_K'-unit equation at a fixed P∈ S_K. The first excludes all non-trivial solutions in K³ when v_P(A)=v_P(B)=m and v_P(C)=m+k, with k>4v_P(2). The second, under an explicit nonvanishing condition on the coefficient valuations, excludes solutions (a,b,c)∈(K^×)³ whose P-valuations are not all equal. In particular, it excludes non-trivial primitive integral solutions with P| abc. Neither criterion restricts the residue degree of P. Both criteria hold unconditionally over every totally real field; over general number fields, we assume residual modularity and Eichler--Shimura-type conjectures. The first criterion covers a new coefficient range. In particular, under the corresponding S_K-unit bound, x^p+y^p=2^rz^p has no non-trivial solutions in K³ for each fixed r≥5 and sufficiently large prime p. This bound holds for a set of real quadratic fields of relative density $1$. For a family of imaginary quadratic fields of relative density 5/6, we explicitly determine the S_K-unit solutions and verify the valuation bound for both criteria when A,B,C∈{±2^r:r∈ℤ_≥0}, including the same r≥5 conclusion.

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BibTeX RIS

BibTeX

@misc{sahoo2026generalized,
  title = {{Generalized Fermat equation over number fields}},
  author = {Satyabrat Sahoo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.28117},
  eprint = {2608.28117},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28117},
}

RIS

TY  - UNPB
TI  - Generalized Fermat equation over number fields
AU  - Satyabrat Sahoo
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.28117
UR  - https://arxiv.org/abs/2608.28117
AB  - Let K be a number field and A,B,C∈𝒪_K∖{0}. Put S_K={P:P|2} and S_K'={q:q|2ABC}. We establish two criteria for the non-existence of non-trivial solutions of Ax^p+By^p=Cz^p over K for sufficiently large prime exponents p. Both criteria use a valuation bound on the S_K'-unit equation at a fixed P∈ S_K. The first excludes all non-trivial solutions in K^3 when v_P(A)=v_P(B)=m and v_P(C)=m+k, with k>4v_P(2). The second, under an explicit nonvanishing condition on the coefficient valuations, excludes solutions (a,b,c)∈(K^×)^3 whose P-valuations are not all equal. In particular, it excludes non-trivial primitive integral solutions with P| abc. Neither criterion restricts the residue degree of P. Both criteria hold unconditionally over every totally real field; over general number fields, we assume residual modularity and Eichler--Shimura-type conjectures. The first criterion covers a new coefficient range. In particular, under the corresponding S_K-unit bound, x^p+y^p=2^rz^p has no non-trivial solutions in K^3 for each fixed r≥5 and sufficiently large prime p. This bound holds for a set of real quadratic fields of relative density $1$. For a family of imaginary quadratic fields of relative density 5/6, we explicitly determine the S_K-unit solutions and verify the valuation bound for both criteria when A,B,C∈{±2^r:r∈ℤ_≥0}, including the same r≥5 conclusion.
ER  -

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