We can similarly use Z[i] to complete the proof, begun in the previous exercise sets, that x = 0, y = 1 is the only integer solution of y³ = x²+1. 7.3.1 Use congruence mod 4 to show that x is even in any integer solution of y3 = x²+1. From now on assume that (x,y) is such a solution. 7.3.2 Explain why gcd(x-i,x+i) = gcd(x+i,2) and use Question 7.3.1 to show that norm(x+i) is odd. 7.3.3 Deduce from Question 7.3.2 that ged(x-i,x+i) = 1. 7.3.4 Deduce, from the previous exercises and unique prime factorization in Z[i], that the factors on the right-hand side of y³ = (x - i)(x+i) are cubes in Z[i]. Likewise, we can find gcd (X - i,X + i) when X is odd, and hence complete the solution of y³ = x² +4 when x = 2X.