Related papers: Complete Padovan sequences in finite fields

Eick & Leedham-Green sketched a construction for infinite sequences of finite $p$-groups with fixed coclass. These infinite sequences have turned out to be very useful in the theory of finite $p$-groups. We exhibit a detailed description…

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and…

In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0,…

Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen…

We present a method for obtaining congruences modulo powers of a prime number~$p$ for combinatorial sequences whose generating function satisfies an algebraic differential equation. This method generalises the one by Kauers and the authors…

We consider the infinite one-sided sequence over alphabet $\{a,b\}$ generated by the period-doubling substitution $\sigma(a)=ab$ and $\sigma(b)=aa$, denoted by $\mathbb{D}$. Let $r_p(\omega)$ be the $p$-th return word of factor $\omega$.…

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that…

We denote the number of distinct topologies which can be defined on a set $X$ with $n$ elements by $T(n)$. Similarly, $T_0(n)$ denotes the number of distinct $T_0$ topologies on the set $X$. In the present paper, we prove that for any prime…

We investigate the linear complexities of the periodic 0-1 infinite sequences in which the periods are the sequence of the parities of the spacings between quadratic residues modulo a prime p, and the sequence of the parities of the…

Shoenfield's completeness theorem (1959) states that every true first order arithmetical sentence has a recursive $\omega$-proof encodable by using recursive applications of the $\omega$-rule. For a suitable encoding of Gentzen style…

A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed…

A real-valued sequence $f = \{ f(n) \}_{n \in \mathbb{N}}$ is said to be second-order holonomic if it satisfies a linear recurrence $f (n + 2) = P (n) f (n + 1) + Q (n) f (n)$ for all sufficiently large $n$, where $P, Q \in \mathbb{R}(x)$…

We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some…

For $n<\omega$, we say that the $\Pi^1_n$-reflection principle holds at $\kappa$ and write $\text{Refl}_n(\kappa)$ if and only if $\kappa$ is a $\Pi^1_n$-indescribable cardinal and every $\Pi^1_n$-indescribable subset of $\kappa$ has a…

Let $\BZ_p$ be the finite field of prime order $p$ and $A$ be a subset of $\BZ_p$. We prove several sharp results about the following two basic questions: (1) When can one represent zero as a sum of distinct elements of $A$ ? (2) When can…

Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call…

We give a full description of all sets of functions on the group $(\mathbb{ Z}_p, +)$ of prime order which are closed under the composition with the clone generated by $+$ from both sides. Thereby, we also get a description of all iterative…

In this paper we obtain some congruences involving central binomial coefficients and Lucas sequences. For example, we show that if p>5 is a prime then $\sum_{k=0}^{p-1}F_k*binom(2k,k)/12^k$ is congruent to 0,1,-1 modulo p according as p=1,4…

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs…

Two infinite 0-1 sequences are called compatible when it is possible to cast out 0's from both in such a way that they become complementary to each other. Answering a question of Peter Winkler, we show that if the two 0-1-sequences are…