Surprises come in different flavors . . . so when I see something that looks like a digit-pattern, I’m surprised.
If you need to see everything in terms of digits, then why not read the original post as inherently about digit patterns? The number 3 can also be written 3.000000000...etc, with a pattern of repeating zeros that never ends. The other integral value I mentioned, sqrt(3)*log(8)/pi = 1.1464562082685322958... doesn't repeat at all. This is more typical of what I would expect given the constraints. Here is another tendril about changing parameters: Integrate[Hypergeometric2F1[1/4,3/4,1,x^2], {x,0,1}]==(1/2)*(ArcCosh[17]+4*ArcSinh[1])/Pi N[% /. Equal -> List, 20] Integrate[Hypergeometric2F1[1/4,3/4,1,1-x^2], {x,0,1}]==Sqrt[2] N[% /. Equal -> List, 20] Neither of these have discernible decimal digit patterns. However sqrt(2) does have a repeating continued fraction, so if you are looking for digit patterns, there you go, another easy one. It's easy to say that you have "scabs of theorems", but I can't make any reasonable response to a statement with no particular meaning. Does one of your "scats of theorems" have anything to do with Ramanujan's elliptic integrals K1, K2, and K3? --Brad