The Sequence That Has a Periodic Table

Start with 1. Read it aloud: "one 1." Write 11. Read 11: "two 1s." Write 21. Then 1211, 111221, 312211. Each term describes the digits of the previous one. John Conway noticed something strange about this sequence in the early 1980s. Each term is roughly 30% longer than the last, and the ratio converges to 1.303577269..., now called Conway's constant. It's an algebraic number of degree 71, the largest real root of a polynomial Conway worked out by hand.

The deeper result is the Cosmological Theorem. No matter what digit string you start with (using 1s, 2s, and 3s), the sequence eventually splits into pieces that never interact across the boundary. There are exactly 92 of these stable pieces. Conway named them after the chemical elements, hydrogen through uranium. There's a sequence called "thulium," one called "plutonium," one called "polonium." Each decays into a specific combination of others: "uranium" splits into "hydrogen" plus "protactinium," and so on.

The constant 1.303577... is the largest eigenvalue of the 92-by-92 matrix describing this decay. The sequence's growth rate is dictated entirely by how those 92 atomic strings transmute into one another. Run the rule long enough on any seed and you get a soup of the same 92 fragments in the same proportions.

Conway's original proof was, by his own admission, lost — never written out in full. The first complete proof on the record came in 1997, when Shalosh B. Ekhad and Doron Zeilberger published "A 2-Minute Proof of Conway's Lost Cosmological Theorem" in the Electronic Research Announcements of the AMS. Shalosh B. Ekhad is Zeilberger's computer. The proof checks the decay table by exhaustive case analysis a machine can finish in seconds — a fitting epitaph for a theorem about a sequence that exists only to be read aloud.