Hey Bruno. I know you wanted to learn about elliptic curves so here's a couple questions. The first doesn't require any knowledge of elliptic curves. Prove that the only integer solutions of y^2 =x^3-2 are (3, plus or minus 5). I was given a similar although more difficult question on an assignment in my algebraic number theory course but I can't find it anywhere so this will have to do.
The second is apparently very difficult (according to the book it is in) and I haven't tried it yet but I'm going to try it tonight so hopefully I'll have the solution before you.
On the curve y^2+y = x^3-7x+6 show that an integral combination a(0,2)+b(1,0)+c(2,0)=0 only for a=b=c=0. So this question requires that you know the point multiplication formula for elliptic curves.
Good luck.
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