Related papers: Narayana Sequences for Cryptographic Applications
We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary…
In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers N_{n,k} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana…
Let p and r be two primes and n, m be two distinct divisors of pr. Consider the n-th and m-th cyclotomic polynomials. In this paper, we present lower and upper bounds for the coefficients of the inverse of one of them modulo the other one.…
We had recently shown that every positive integer can be represented uniquely using a recurrence sequence, when certain restrictions on the digit strings are satisfied. We present the details of how such representations can be used to build…
Non-periodic tilings of $\mathbb{Z}_N$ are a difficult to obtain key to a wide range of complex mathematical issues. In a musician's eyes, they are called rhythmic Vuza canons. After a short summary of the main properties of rhythmic tiling…
Parallel supercomputer-based Monte Carlo applications depend on pseudorandom number generators that produce independent pseudorandom streams across many separate processes. We propose a new scalable class of parallel pseudorandom number…
Many famous integer sequences including the Catalan numbers and the Motzkin numbers can be expressed in the form $ConstantTermOf\left[P(x)^nQ(x)\right]$ for Laurent polynomials $Q$, and symmetric Laurent trinomials $P$. In this paper we…
We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the B\'ezout number of the system and linear in its bit size. Our…
We review the concept of the number wall as an alternative to the traditional linear complexity profile (LCP), and sketch the relationship to other topics such as linear feedback shift-register (LFSR) and context-free Lindenmayer (D0L)…
The generalized cyclotomic binary sequences $S=S(a, b, c)$ with period $n=pq$ have good autocorrelation property where $(a, b, c)\in \{0, 1\}^3$ and $p, q$ are distinct odd primes. For some cases, the sequences $S$ have ideal or optimal…
We consider a p-wave superconductor wire coupled to two metallic rings. confined to a one-dimensional wire. At the two interface between the the wire and the metallic rings the pairing order parameter vanishes, as result two zero modes…
In this paper, we shall find a new connection between $n$th degree polynomial mod $p$ congruence with $n$ roots and higher-order Fibonacci and Lucas sequences. We shall first discuss the recent work been done in sequences and their…
The Fibonacci sequence is periodic modulo every positive integer $m>1$, and perhaps more surprisingly, each period has exactly 1, 2, or 4 zeros that are evenly spaced, which also holds true for more general $K$-Fibonacci sequences. This…
Prime reciprocals have applications in coding and cryptography and for generation of random sequences. This paper investigates the structural redundancy of prime reciprocals in base 10 in a manner that parallels an earlier study for binary…
The paper exploits an isomorphism between the natural numbers N and a space U of periodic sequences of the roots of unity in constructing a recursive procedure for representing and computing the prime numbers. The nth wave number ${\bf…
Let n be a positive integer and let p be a prime. We calculate the probability that a random monic polynomial of degree n with coefficients in the ring Z_p of p-adic integers splits over Z_p into linear factors.
In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of…
The purpose of this note is to obtain some congruences modulo a power of a prime $p$ involving the truncated hypergeometric series $$\sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a}$$ for $a=1$ and $a=2$. In the last section,…
In this paper, we focus on analyzing the period distribution of the inversive pseudorandom number generators (IPRNGs) over finite field $({\rm Z}_{N},+,\times)$, where $N>3$ is a prime. The sequences generated by the IPRNGs are transformed…
We prove that if the signed binomial coefficient $(-1)^i\binom{k}{i}$ viewed modulo p is a periodic function of i with period h prime to p in the range $0\le i\le k$, then k+1 is a power of p, provided h is not too large compared to k. (In…