In honor of pi day (^{3}⁄_{14})…

If pi has an infinite number of digits in random sequence, then pi must contain all sequences.

It is commonly said that “pi does not repeat” and while generally true (pi does not predictably repeat) I think this assertion is clearly not strictly correct – there are an infinite number of repeated sequences in pi (e.g. “123123” occurs in the first million digits) as subsets of the sets of numbers in pi.

Therefore, I postulate that there exists at least one decimal position N in pi that is followed by the exact sequence of digits 1 to N-1. Namely, pi is “3.14159…314159..”. In regular expression syntax this question would actually just be phrased as /^([0-9]+){2}/, omitting the period after 3, naturally.

It should be provable that not only such an N exists but there is an infinite set of such repeating points. I would guess that that first N would be very large, maybe larger than the number of digits of pi yet computed (in the trillions), but if there is one there are almost assuredly an infinite number of other repeating points. This set would start with an almost improbably large number and I would suppose the numbers would very quickly get ridiculously larger.

There would be different such sets for different transcendentals, so perhaps we can discuss the creation of a function “D” that defines the infinite set of these “repeating points” for any transcendental. D(π) , D(e), D(φ), ~~D(√2)~~, etc.