Baltic Way 2011 · Shortlist problem

Number Theory

Find all integer solutions to the equation

2x^3 - y^2 = 3.

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Topics

Diophantine equations · Modular arithmetic

Solutions

Solution

Consider the equation modulo $8$ . Then $y^2 \equiv 0, 1, 4 \pmod{8}$ . This means $2x^3 \equiv 3, 4, 7 \pmod{8}$ all of which are impossible. The equation lacks a solution.