On the Non–Negative Integer Solution of Diophantine Equation x2 + x + 1 = 11y

Abstract

Diophantine equations have a vast field of applications to solve the problems of astronomy, algebra, trigonometry, cryptography, geometry, and chemistry. These equations can address difficult and numerous problems in number theory and shed light on the distribution of prime numbers since they look for integer solutions to polynomial equations. They aid in the determination of celestial mechanics and orbital parameters in astronomy. They also aid in the comprehension of form characteristics and the connections between various figures in algebra and geometry. Diophantine equations are used in cryptography to provide safe communication channels that guarantee data secrecy and integrity. They may also be used to investigate molecular structures and balance equations in chemistry. These equations need special attention to solve because there is no universal technique for such equations. In this paper, authors examined the Diophantine equation , , where  represents the set of non-negative integers, for determining the ordered pairs  that satisfy the equation . For this purpose, authors have considered the well known modular arithmetic technique. Results show that the ordered pair  is the only solution of the Diophantine equation .