If you require that a JAPH must be no more than 4 lines of no more than 80 characters each, then there are, at most, 128 (different ASCII characters) ** 320 different ways to write 320 characters (That's equivalent to 2 ** 2240). Since not all JAPHs are exactly 320, the actual maximum limit is the sum of 128 to the N where N varies from 1 to 320. Or, 127 * SUM( 128**N, N:0->319 ) (or SUM(2**(7N+1),N:0-319) - 1). Of course, you can start chipping away at that pretty quickly. I think the shortest JAPH is something like 50 characters, so that becomes your lowest exponent instead of 1. Not all the ASCII characters are usable, and a lot of them are interchangeable in many cases ($x vs. $y, etc), so that drops the 128 quite a bit. Factor in some more fancy math and I suspect the number drops to something like 2 ** 200 (or so). Or, roughly, 10 ** 66-ish.
In other words, if every possible JAPH were written, one per second, from now onwards, you'd run out of hydrogen in the Sun long before running out of JAPHs. Or, put it another way, if you were to write one JAPH per second from the Big Bang onwards, you'd run out of Universe before you run out of JAPHs.
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