The digits of pi have never repeated and never will

Pi is irrational, so its decimal expansion goes on forever without settling into a repeating pattern. It has been computed to trillions of digits. Pi is irrational, meaning it cannot be written as one whole number divided by another. That property is exactly what guarantees its decimal expansion never terminates and never settles into a repeating block, however far out you compute it. Every fraction eventually repeats — a seventh gives 142857 forever — so proving pi is not a fraction proves the digits never cycle. Lambert established this in 1761. A separate and much harder question is whether pi is normal: whether every digit and every sequence of digits appears equally often in the long run. It looks that way across the trillions of digits computed, but nobody has proved it. If true, any number you can name — a phone number, a birthday — appears somewhere in the expansion.

The digits of pi have never repeated and never will

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