As the question was about bitcoin, I'm not sure how prime factoring is relevant.
> Can I run Shors algorithm on this? If not, how many qubits would I need for that?
The question was about Shor's algorithm. The question relating to bitcoin seemed like an aside.
https://en.wikipedia.org/wiki/Shor%27s_algorithm
Given that the original article was about Quantum Computing, I was more interested in answering questions about Quantum Computing.
Yes, but bitcoin's authentication is via ECDSA. Elliptic curve discrete log is solvable with Shor's algorithm, but the qubit calculation would be different.
Wikipedia seems to indicate that 2330 qubits and 126 billion Toffoli gates would be required for elliptic curves with 256-bit group size, although I wonder if the endomorphism of secp256k1 gives an additional speedup.
https://arxiv.org/abs/quant-ph/0205095
You can implement Shor's algorithm to factor an N-bit number with just 2n+3 qubits. So the 20 qubit processor would be able to factor any 8 bit number (not necessarily faster than a classical computer could, or even a human could, mind you).