I'm working on a previous exam problem, and my solution does not match with the given one, and I don't know why.
I have the elliptic curve $$E: Y^{2} = X^{3} + X + 46$$ over $\mathbb{F_{101}}$. We're also given the congruences $$N\equiv 1 (mod 3)$$ $$N\equiv 4 (mod 5)$$ as well as the information that $N$ is prime.
I easily found that $N = 4 (mod 15)$ will solve this, and then I have a look at Hasses' Theorem in my Crypto. book: $$(p + 1)-2\sqrt{p} \leq N \leq (p + 1) + 2\sqrt{p}$$ where $p = 101$. Somehow I end up with the range $82 \leq N \leq 122$ when the solution should be $89 \leq N \leq 130$, thus giving $N=109$. Of course, with my range, I still find $N=109$ to be the only possible solution as it is the only prime which is congruent to $4$ mod $15$. But it bugs me that my range is different from that in the given solution. Have I misunderstood something here? It's been some time since I've actually solved congruences, maybe I messed up there.