Are there general methods / algorithms for finding a Turing Machine that will output a given binary number?
For example, I want the machine to write 0111001101111001110110001110110011000010110100101101110 on a blank tape and then halt. Is there a general way to find a machine that does that?
I could try random machines, or enumerate all 10-state machines, or something, but that would be ridiculously slow, so I'm wondering whether/where the idea has been explored. I'm asking here because I frankly don't even know what keywords to search for on the googles.
Background : I just think it would be a cool form of extreme compression, with no practical purpose but recreation.