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That makes me wonder if it's also easy to create a string that is its own MD5.

This is called a fixed point, that is an X such that md5(X)==X. A quick google for "md5 fixed point" shows some claims that the probability that one exists is about 63% (https://news.ycombinator.com/item?id=614027) but no examples of a known one

Also the "real" fixed point wouldn't be necessarily ASCII

So you could have two types: one where the hash matches the input bytes, another where an hex string is inputed and that matches the hex representation of the hash

Also EBCDIC and UTF-16

There is http://www.olegkikin.com/md5game/ on which I wasted some GPU cycles many years ago. It uses a very silly metric for progress though. Finding a longer substring that matches doesn't bring you closer to finding MD5(x) = x. It was a fun problem to try out some crypto bruteforcing on the GPU though.

I don't think any of the known weaknesses of MD5 would help you in finding MD5(x) = x.

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