On the Diophantine Equation 𝒓𝟑 − 𝟒𝒓𝟐 + 𝟑𝒓 + 𝟏 = 𝟓s

Abstract

In this paper, authors have examined the Diophantine equation 𝑟³ − 4𝑟² +3𝑟+1= 5^𝑠, 𝑟,𝑠 ∈ 𝑍0, where 𝑍0 represents the set of non negative integers, for determining the ordered pairs (𝑟,𝑠) ∈ 𝑍0 × 𝑍0 that satisfy the equation 𝑟³ − 4𝑟² + 3𝑟 + 1 = 5^𝑠. For this purpose, authors have considered the well known modular arithmetic technique. It was shown by the results of this paper that the ordered pairs (𝑟,𝑠) = (0,0),(1,0),(3,0) ∈ 𝑍0 × 𝑍0 are the only solutions of the Diophantine equation 𝑟³ − 4𝑟² + 3𝑟 + 1 = 5^𝑠.