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.

On the Diophantine Equation (p+12)x+(p+2k)y=z2 where p is a Prime Number

“…In the same year, he proved that the Diophantine equation ‫‬ ௫ + ሺ‫‬ + 5ሻ ௬ = ‫ݖ‬ ଶ when p is prime where ‫‬ + 5 = 2 ଶ௨ has no solution (x, y, z) in positive integers and proved that the solution to the Diophantine equation ‫‬ ௫ + ‫ݍ‬ ௬ = ‫ݖ‬ ଷ when ≥ 2 , q are primes, 1 ≤ ‫,ݔ‬ ‫ݕ‬ ≤ 2 are integers. In 2021, Moonchaisook [11] proved that the non-linear Diophantine equation ‫‬ ௫ + ሺ‫‬ + 4 ሻ ௬ = ‫ݖ‬ ଶ has no solution, when ‫‬ > 3, ‫‬ + 4 are primes numbers, when x, y and z are non-negative integers and n is positive integer, this year Aggarwal [1] (2021) studied solutions to the exponential Diophantine equation ሺ2 ଶାଵ − 1ሻ + 13 = ‫ݖ‬ ଶ where m, n are whole numbers, has no solution in whole number.…”