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

Abstract

In this paper, authors have examined the Diophantine equation 𑟳 − 4𑟲 +3ð‘Ÿ+1= 5^ð‘ , ð‘Ÿ,𑠠∈ ð‘0, where ð‘0 represents the set of non negative integers, for determining the ordered pairs (ð‘Ÿ,ð‘ ) ∈ ð‘0 × ð‘0 that satisfy the equation 𑟳 − 4𑟲 + 3ð‘Ÿ + 1 = 5^ð‘ . 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 (ð‘Ÿ,ð‘ ) = (0,0),(1,0),(3,0) ∈ ð‘0 × ð‘0 are the only solutions of the Diophantine equation 𑟳 − 4𑟲 + 3ð‘Ÿ + 1 = 5^ð‘ .