The Fermat-type equation with signature (2, 2, n) and Bunyakovsky conjecture

Abstract

We first discuss the Fermat-type equation with signature (2, 𝑚, 𝑛), which is the Diophantine equation in the shape 𝑥 2 + 𝑦 𝑚 = 𝑧 𝑛 , where 𝑥, 𝑦 and 𝑧 are unknown integers, and 𝑚, 𝑛 are fixed positive integers greater than 1. This kind of equations has been particularly focused on our work here in the case 𝑚 = 2, 𝑛 = 5 and 𝑦 = 𝑝 is a fixed rational prime. Then the first result describing the condition of such a prime 𝑝 in order to illustrate that this certain equation has an integer solution (𝑥, 𝑦) when 𝑝 ≡ 1(mod 4) and gcd(𝑥, 𝑝) = 1, and the second result stating that the equation has no integer solution (𝑥, 𝑦) when 𝑝 ≡ 3(mod 4) are provided. Lastly, we will indicate that the results of Be ́rczes and Pink about solving the equation 𝑥 2 + 𝑝 2𝑘 = 𝑧 𝑛 in 2008 have been generalized in the particular cases (𝑛, 𝑘) = (3,1) and (5,1), and additionally present that the first result and also its analogous result in the particular case 𝑛 = 3 can be linked to the Bunyakovsky conjecture.

Keywords

• fermat- type equations
• gaussian integers
• unique factorization
• bunyakovsky conjecture