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).)
No such word exists over any alphabet of 3 or fewer integers, and the question is open for alphabets of size 5. This site tracks, for each alphabet, the longest known finite word avoiding additive squares — and, where an exhaustive search has settled it, the exact maximum.
Records
One point per canonical alphabet: the longest verified additive-square-free word against the alphabet’s diameter (max letter − min letter). Squares are closed alphabets whose exact maximum is known. The dashed line is an empirical guide, not a theorem. Browse the full table →
| alphabet | size | diameter | record | status |
|---|---|---|---|---|
| {0, 2, 7, 21, 36} | 5 | 36 | 6212 | open |
| {0, 1, 5, 11, 22} | 5 | 22 | 1997 | open |
| {0, 1, 4, 9, 18} | 5 | 18 | 1506 | open |
| {0, 1, 3, 7, 15} | 5 | 15 | 896 | open |
| {0, 1, 3, 7, 12} | 5 | 12 | 607 | open |
| {0, 1, 2, 4, 8} | 5 | 8 | 331 | open |
| {0, 1, 2, 4, 7} | 5 | 7 | 317 | open |
| {0, 1, 2, 4, 5} | 5 | 5 | 148 | open |
| {0, 1, 2, 3, 5} | 5 | 5 | 144 | open |
| {0, 1, 3, 7} | 4 | 7 | 106 | open |
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.