If you have two valid signatures using the same private key where k' = 2k then:
From each message we can derive the z value (hash of the message) so:
First message and signature (m, r, s, z)
Second message and signature (m', r', s', z')
Therefore: ks = z + rdA and k's' = z' + r'dA
Therefore: (sk - z)/r = (s'k' - z')/r'
But in this case k' = 2k so:
(sk - z)/r = (2s'k - z')/r'
So all you have to do is solve for k. All the other values: s, z, r, s', z', and r' are all known.
rr'(sk - z)/r = rr'(2s'k - z')/r'
r'(sk - z) = r(2s'k - z')
r'sk - r'z = 2rs'k - rz'
r'sk - r'z +rz' = 2rs'k
k = (r'sk - r'z +rz')/2rs'
Once you know k you can simply calculate the private key, dA = (sk - z)/r
I still do not see what your two signatures have to do with the BTC at 1FvUkW8thcqG6HP7gAvAjcR52fR7CYodBx
These two things: how to solve for the private key when you know k' = 2k and the BTC stored at 1FvUkW8thcqG6HP7gAvAjcR52fR7CYodBx seem to be unrelated, right?
That would be too easy! I reversed the order between s and r of the second signature ... You just need an efficient equation and the private key is all yours. Anyone who can calculate will own the 3,500.00 BTCs
You don't need to post formulas or equations here, just calculate and get the private key for yourself.