Are there any recurring patterns in pi?

Answer 1

There are no recurring patterns since pi is an irrational number. If there was a recurruing pattern, pi would be a rational number (it could be expressed exactly as a fraction).

If you mean are there any repeated patterns in pi, logically there have to be.

Take any digit - let's say 3. Whenever 3 occurs in the decimal expansion of pi it must be followed by another digit. There are ten possible digits, and when these have been exhausted, one of them must be repeated. So there will be a two-digit repeted pattern. Because pi never terminates, any given two-digit patterns will occur again and again and must be followed by another digit each time, so there will be three-digit repeated patterns. And so on and on.

There will (eventually) be repeated patterns of any length you choose - repeated hundred-digit patterns, or thousand-digit ones, or million-digit ones. You'll have to search a long, long way to find them, though! In the first 4 billion digits of pi there don't appear to be any repeating patterns longer than 10 digits.