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    Mordell Equations

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    Exercise 5. The aim of this exercise is to prove that the Mordell equation y^2 = x^3 - 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 y^2 + 4 = (x-1)(x^2 + x +1)
    3. Show that x^2 + x + 1 is congruent to 3 mod 4 and that x^2 + x + 1 greater than 3
    4. Prove that x^2 + x + 1 has a prime factor p congruent to 3 mod 4

    See attachment for remainder of problem set.

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    Solution Preview

    1. Assuming there is a solution, we reduce the equation to modulo 4.
    We tabulate the values of x and y as ...

    Solution Summary

    In this solution, a particular situation of Mordell equation is discussed with steps given to solve the problem set.