The aim of this exercise is to prove that the mordell

Question

The aim of this exercise is to prove that the Mordell equation y² = x³ - 5 has no solutions. We proceed by contradiction and assume that (x, y) is an integral solution.

1. By reducing Mordell equation mod 4, show that y is even and x =- 1 mod 4.

2. Show that y² + 4 = (x - 1)(x² + x + 1).

3. Show that x² + x +1 is congruent to 3 mod 4 and that x² + x +1 ≥ 3.

4. Prove that x² + x 1 has a prime factor p congruent to 3 mod 4.

5. Prove that -4 ∈ S_p and then that -1 ∈ S_p.

6. Conclude that the Mordell equation y² = x³ - 5 has no solutions.