the largest integer that has been factored using Shor's algorithm on a quantum computer is 21. In 2019, a team of researchers at Google successfully factored the number 21 using their quantum computer known as "Sycamore."
However, it's important to note that Shor's algorithm's main application is in factoring large composite numbers into their prime factors, which has implications for cryptography. Currently, classical computers are much more efficient in factoring large numbers compared to quantum computers. The largest numbers factored using classical methods are significantly larger than what has been achieved with quantum computers.
Please note that advancements in quantum computing are rapid, and there may have been further progress since I'm not interested in that topic as much as i used to and my answer might be a bit outdated. I recommend checking for the latest research and developments in the field to get the most up-to-date information.