相关论文: Cubefree binary words avoiding long squares
Recently, Grytczuk, Kordulewski, and Niewiadomski defined an extremal word over an alphabet $\mathbb{A}$ to be a word with the property that inserting any letter from $\mathbb{A}$ at any position in the word yields a given pattern. In this…
The number of square-free integers in $x$ consecutive values of any polynomial $f$ is conjectured to be $c_fx$, where the constant $c_f$ depends only on the polynomial $f$. This has been proven for degrees less or equal to 3. Granville was…
We establish Schmutz Schaller's conjecture that the hexagonal lattice is `better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in…
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $n^2+1$, $n^2+2$. We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given…
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[\alpha n]$, $[\alpha n]+1$, where $\alpha>1$ is irrational number with bounded partial quotient or irrational algebraic number.
An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the…
Given a negative $D>-(\log X)^{\log 2-\delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of…
We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered…
In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f= (p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern…
A conjecture of Chung and Graham states that every $K_4$-free graph on $n$ vertices contains a vertex set of size $\lfloor n/2 \rfloor$ that spans at most $n^2/18$ edges. We make the first step toward this conjecture by showing that it…
We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli.
A long-standing conjecture states that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, 364, 420, 428, 454 is a sum of at most seven positive cubes. This was first observed by Jacobi in 1851 on…
A well known conjecture asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ can not be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the geometry of cubic fourfolds is intricately…
A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the…
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…
A conjecture of M\'a\u{c}ajov\'a and \u{S}koviera asserts that every bridgeless cubic graph has two perfect matchings whose intersection does not contain any odd edge cut. We prove this conjecture for graphs with few vertices and we give a…
We introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the…
In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $x^2+y^2+1$, $x^2+y^2+2$. We also give an asymptotic formula for the number of pairs of positive integers $x, y \leq H$ such that…
We determine all modular curves $X_0(N)/\langle w_d\rangle$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$, when $N$ is square-free.
A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: For each pair of positive integers $(n,k)$ with $3\le k\le n-3$, there is a $k$-dimensional $\Bbb F_2$-subspace $E$…