Assignment:
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 greater than 3
4. Prove that x2 + x + 1 has a prime factor p congruent to 3 mod 4
5. Provwe that (-4)‾ ∈ Sp and then that (-1)‾ ∈ Sp.
6. Conclude that the Mordell equation y2 = x3 - 5 has no solutions.
Provide complete and step by step solution for the question and show calculations and use formulas.