On the Non-Linear Diophantine Equation 𝒑 𝒙 + (𝒑 + 𝟒 𝒏) 𝒚 = 𝒛 𝟐 where p and 𝒑 + 𝟒 𝒏 are Primes

Abstract

In this paper, we consider the non-linear Diophantine equation 𝑝 𝑥 + (𝑝 + 4 𝑛) 𝑦 = 𝑧 2 , where 𝑝 > 3, 𝑝 + 4 𝑛 are primes, x, y and z are nonnegative integers and n is a natural number. It is shown that this non-linear Diophantine equation has no solution.