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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 ∈ Sp and then that -1 ∈ Sp.
6. Conclude that the Mordell equation y2 = x3 - 5 has no solutions.
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