How to show there exists no solution to a discrete logarithm problem on an Elliptic Curve? The exact problem is to show that $\nexists$k such that $k(1,2) = (4,5)$ on the elliptic curve defined by $\widetilde{E}: y^2 = x^3 -14x + 17$ over $\mathbb Q$.
Background:
E: $y^2 = x^3 + 3$ over $F_7$. On E we have 4(1,2) = (4,5).
Now the points P = (1,2) and Q = (4,5) are on both E and $\widetilde{E}$ and $\widetilde{E}$ was found by uplifting E (from mod 7 to Q). Also it was discovered that 2(1,2) = (1,-2) and 3(1,2) = $\infty $ mod 73 on $\widetilde{E}$.

You have found that modulo $73$, $(1,2)$ is a point of order $3$.
Since $(4,5)$ modulo $73$ is neither $(1,2),(1,-2)$ nor $\infty$, it can't be in the group generated by $(1,2)$.
You should try to compute the group of the curve reduced modulo a few primes and find primes  other than $73$ where the reduction of $(4,5)$ is not in the subgroup generated by the reduction of $(1,2)$.
