Two results on 𝑥^{𝑟}+𝑦^{𝑟}=𝑑𝑧^{𝑝}
- 7 November 2023
- journal article
- research article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 152 (02) , 591-598
Abstract
This note proves two theorems regarding Fermat-type equation x r + y r = d z p x^r + y^r = dz^p where r ≥ 5 r \geq 5 is a prime. Our main result shows that, for infinitely many integers d d , the previous equation has no non-trivial primitive solutions such that 2 ∣ x + y 2 \mid x+y or r ∣ x + y r \mid x+y , for a set of exponents p p of positive density. We use the modular method with a symplectic argument to prove this result.Keywords
Funding Information
- Consejo Superior de Investigaciones CientÃficas (IP-2018-01-1313)
- Consejo Superior de Investigaciones CientÃficas (Grant KK.01.1.1.01.0004)
- Hrvatska Zaklada za Znanost (IP-2018-01-1313)
- Hrvatska Zaklada za Znanost (Grant KK.01.1.1.01.0004)
- European Regional Development Fund (IP-2018-01-1313)
- European Regional Development Fund (Grant KK.01.1.1.01.0004)
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