A New Approach to Find Integral Solutions for the Mordell’s Equation, 𝑦 2 + 2 = 𝑥 3

Abstract

— Mordell’s equation, 𝑦 2 + 2 = 𝑥 3 , which is historically important, was solved using complex numbers and more specifically using the unique factorization method. In this paper, it is shown that Mordell’s equation can be solved by using elementary mathematics and the Fermat’s little theorem. In the first step, it is shown that if (𝑥, 𝑦) is a solution of the aforementioned equation then 𝑥 ≠ 𝑦 and then the equation is reduced to a cubic equation. In the next step, it is shown that this cubic equation has no other integer solution than 𝑥 = 3 using very elementary mathematics and the Fermat’s little theorem, and hence the Mordell’s equation has only the well-known solution 𝑥 = 3, 𝑦 = ±5.