In this study, we determine when the Diophantine equation $x^2-kxy+y^2-2^n=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\le n\le 10.$ Moreover, we give all positive integer solutions of the same equation for $0\le n\le 10$ in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation $x^2-kxy+y^2-2^n=0$.

A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character sums.

In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the $L_1$-norm of trigonometric sums.

Let G be a finite cyclic group. Every sequence S over G can be written in the form $S=\left(n₁g\right)·...·\left(n_{l}g\right)$ where g ∈ G and $n₁,...,n_{l}i\in [1,ord\left(g\right)]$, and the index ind(S) is defined to be the minimum of $(n₁+\cdots +n_l)/ord\left(g\right)$ over all possible g ∈ G such that ⟨g⟩ = G. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|,6) = 1 has index 1. This conjecture was confirmed recently for the case when |G| is a product of at most two prime powers. However, the general case is still open. In this paper, we make some...

For each $m\ge 2,$ we consider the $m$-bonacci numbers defined by $F_k=2^k$ for $0\le k\le m-1$ and $F_k=F_{k-1}+F_{k-2}+\cdots +F_{k-m}$ for $k\ge m.$ When $m=2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$-bonacci numbers in one or more different ways. Let $R_m\left(n\right)$ be the number of partitions of $n$ as a sum of distinct $m$-bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_m\left(n\right)$ involving sums of binomial coefficients modulo $2.$ In addition we show that this formula may be used to determine the number of partitions...

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general...

We study the structure of longest sequences in $\mathbb{Z}{ₙ}^d$ which have no zero-sum subsequence of length n (or less). We prove, among other results, that for $n=2^a$ and d arbitrary, or $n=3^a$ and d = 3, every sequence of c(n,d)(n-1) elements in $\mathbb{Z}{ₙ}^d$ which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where $c(2^a,d)=2^d$ and $c(3^a,3)=9$.

A graph G is called (H;k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its k vertices. Q(H;k) denotes the minimum size among the sizes of all (H;k)-vertex stable graphs. In this paper we complete the characterization of $(K_{m,n};1)$-vertex stable graphs with minimum size. Namely, we prove that for m ≥ 2 and n ≥ m+2, $Q(K_{m,n};1)=mn+m+n$ and $K_{m,n}*K₁$ as well as $K_{m+1,n+1}-e$ are the only $(K_{m,n};1)$-vertex stable graphs with minimum size, confirming the conjecture of Dudek and Zwonek.

Soit $E$ une courbe elliptique sur $\mathbb{Q}$ par un modèle de Weierstrass généralisé :$$y^2+A_{1}xy+A_{3}y=x^3+A_2{x}^2+A_{4}x+A_6;A_i\in \mathbb{Z}.$$Soit $M=(a/d^2,b/d_3)$ avec $(a,d)=1$, un point rationnel sur cette courbe. Pour tout entier $m$, on exprime les coordonnées de $mM$ sous la forme :$$mM=\left(\frac{{\phi }_m\left(M\right)}{{\psi }_n^2\left(m\right)},\frac{{\omega }_m\left(M\right)}{{\psi }_m^3\left(M\right)}\right)=\left(\frac{{\widehat{\phi }}_m}{d^2{\widehat{\psi }}_m^2},\frac{{\widehat{\omega }}_m}{d^3{\widehat{\psi }}_m^3}\right),$$où ${\phi }_m,\psi \_m,{\omega }_m\in \mathbb{Z}[A_1,\cdots ,A_6,x,y]$ et ${\widehat{\phi }}_m$, ${\widehat{\psi }}_m$, ${\widehat{\omega }}_m$ sont déduits par multiplication par des puissances convenables de $d$.Soit $p$ un nombre premier impair et supposons que $M\left(\mathrm{mod}p\right)$ est non singulier et que le rang d’apparition de $p$ dans la suite d’entiers $\left({\widehat{\psi }}_m\right)$ est supérieur ou égal à trois. Notons ce rang par $r=r\left(p\right)$ et soit ${\nu }_p({\widehat{\psi }}_r)=e_0\ge 1$. Nous montrons que la suite $\left({\widehat{\psi }}_m\right)$...

Soit $c\>1$, $p$ un nombre premier et $𝒜$ une partie de $\mathbb{Z}/p\mathbb{Z}$ de cardinal supérieur à $c\sqrt{p}$ telle que pour tout sous-ensemble non vide $ℬ$ de $𝒜$, on a ${\sum }_{b\in ℬ}b\ne 0$. On montre qu’il existe $s$ premier à $p$ tel que l’ensemble $s.𝒜$ est très concentré autour de l’origine et qu’il est presque entièrement composé d’éléments de partie fractionnaire positive. Plus précisément, on a$$\sum _{a\in 𝒜}∥\frac{sa}{p}∥\<1+O(p^{-1/4}lnp)\text{et}\sum _{\begin{array}{c}a\in 𝒜,\\ \{sa/p\}\ge 1/2\end{array}}∥\frac{sa}{p}∥=O(p^{-1/4}lnp).$$On montre également que les termes d’erreurs ne peuvent être remplacés par $o\left(p^{-1/2}\right)$.