Does a bi-infinite additive-square-free word exist?
An additive square is a pair of adjacent blocks of the same length with
the same sum, like [3 0 4] [2 5 0].
An open problem, posed independently by
Pirillo–Varricchio
(1994) and
Halbeisen–Hungerbühler
(2000), asks:
is there an infinite word over a finite set of integers containing no additive square?
Equivalently: is there a Lipschitz function
f : ℤ → ℤ whose graph contains no nontrivial
three-term arithmetic progression? (The letters are the increments
f(n+1) − f(n).)
The challenge: pick any finite set of integers and write the longest word you can in which no two adjacent equal-length blocks have equal sums. Every alphabet has its own record to beat — and an alphabet whose records never stop growing would settle the problem.
No such word exists over any alphabet of 3 or fewer integers; every 4-letter alphabet settled so far has a finite maximum (14 are closed exactly here); from size 5 up the question is wide open. Progress runs in two directions. Push a record up: every longer word is a better lower bound, and a family of records growing without bound answers yes. Close an alphabet: an exhaustive search certifies its exact maximum, and if every alphabet eventually closes, the answer is no. On the plot below, a record lifts its point higher; a closure turns it into a filled square.
Records
One point per canonical alphabet: the longest verified additive-square-free word against the alphabet’s diameter d (max letter − min letter). Squares are closed alphabets whose exact maximum is known. The dashed line is the 100,000-letter submission cap. Color is the alphabet size:
Submit a word
Paste a word: integers separated by spaces, commas, or line breaks. The word is checked exactly, then normalized — translating, scaling, and reflecting an alphabet gives an equivalent word, so records are kept per canonical alphabet (minimum 0, gcd 1, lexicographically ≤ its mirror). A record is set when your word is strictly longer than the alphabet’s current best. Alphabets need at least 4 distinct letters — smaller ones are settled (additive squares are unavoidable there), so such words are verified but not recorded.