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.© BrainMass Inc. brainmass.com March 5, 2021, 1:33 am ad1c9bdddf
1. Assuming there is a solution, we reduce the equation to modulo 4.
We tabulate the values of x and y as ...
In this solution, a particular situation of Mordell equation is discussed with steps given to solve the problem set.