Related papers: The Simplest Binary Word with Only Three Squares
In this article, we study a sum of squares of integers except for a fixed one. For any nonnegative integer $n$, we find the minimum number of squares of integers except for $n$ whose sums represent all positive integers that are represented…
Earlier work on machine learning for automated reasoning mostly relied on simple, syntactic features combined with sophisticated learning techniques. Using ideas adopted in the software verification community, we propose the investigation…
We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood.…
We give combinatorial proofs for some identities involving binomial sums that have no closed form.
Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an…
An abstract indefinite least squares problem with a quadratic constraint is considered. This is a quadratic programming problem with one quadratic equality constraint, where neither the objective nor the constraint are convex functions.…
In [2], while studying a relevant class of polyominoes that tile the plane by translation, i.e., double square polyominoes, the authors found that their boundary words, encoded by the Freeman chain coding on a four letters alphabet, have…
A binary shuffle square is a binary word of even length that can be partitioned into two disjoint, identical subwords. Huang, Nam, Thaper, and the first author conjectured that as $n\rightarrow \infty$, asymptotically half of all binary…
We consider partial words with a unique position starting a power. We show that over a $k$ letter alphabet, a partial word with a unique position starting a square can contain at most $k$ squares. This is in contrast to full words which can…
We introduce an elementary congruence-based procedure to look for q-th power multiples in arbitrary binary recurrence sequences (q>2). The procedure allows to prove that no such multiples exist in many instances.
We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few non-zero binary digits.
We consider the language consisting of all words such that it is possible to obtain the empty word by iteratively deleting powers. It turns out that in the case of deleting squares in binary words this language is regular, and in the case…
Improved upper and lower bounds on the number of square-free ternary words are obtained. The upper bound is based on the enumeration of square-free ternary words up to length 110. The lower bound is derived by constructing generalised…
Entringer, Jackson, and Schatz conjectured in 1974 that every infinite cubefree binary word contains arbitrarily long squares. In this paper we show this conjecture is false: there exist infinite cubefree binary words avoiding all squares…
A square is a word of the form $xx$ for a non-empty word $x$. Brlek and Li [Comb. Theory, 2025] proved that the number of distinct squares in a word $w$ of length $n$ is at most $n - \sigma$, where $\sigma$ is the number of letters used in…
A sum-of-squares is a polynomial that can be expressed as a sum of squares of other polynomials. Determining if a sum-of-squares decomposition exists for a given polynomial is equivalent to a linear matrix inequality feasibility problem.…
We present a new recursive generation algorithm for prefix normal words. These are binary strings with the property that no substring has more 1s than the prefix of the same length. The new algorithm uses two operations on binary strings,…
We consider integers whose squares have just three decimal digits. Examples are e.g. given by $2108436491907081488939581538^2 = 4445504440405440505004450045555054500055550554550445444$ and $10100000000010401000000000101^2 =…
Linear-time computational techniques have been developed for combining evidence which is available on a number of contending hypotheses. They offer a means of making the computation-intensive calculations involved more efficient in certain…
We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded…