In the three preliminary sections to the article we define two operations on finite sequences which seem to be of general interest. The first is the cut operation that extracts a contiguous chunk of a finite sequence from a position to a position. The second operation is a glueing catenation that given two finite sequences catenates them with removal of the first element of the second sequence....

This paper describes a parallel algorithm and its needed architecture and a complementary sequential algorithm for solving sequence alignment problem on DNA (Deoxyribonucleic acid) molecules. The parallel algorithm is considered much faster than sequential algorithms used to perform sequence alignment; the initialization operation is done by activating a number of processing elements each compa...

The discriminator of an integer sequence s = (s(i))i≥0, introduced by Arnold, Benkoski, and McCabe in 1985, is the map Ds(n) that sends n ≥ 1 to the least positive integer m such that the n numbers s(0), s(1), . . . , s(n − 1) are pairwise incongruent modulo m. In this note we consider the discriminators of a certain class of sequences, the k-regular sequences. We compute the discriminators of ...

Let p be an odd prime and let a, m ∈ Z with a > 0 and p ∤ m. In this paper we determine p a −1 k=0 2k k+d /m k mod p 2 for d = 0, 1; for example, p a −1 k=0 2k k m k ≡ m 2 − 4m p a + m 2 − 4m p a−1 u p−(m 2 −4m p) (mod p 2), where (−) is the Jacobi symbol and {u n } n0 is the Lucas sequence given by u 0 = 0, u 1 = 1 and u n+1 = (m − 2)u n − u n−1 (n = 1, 2, 3,. . .). As an application, we deter...

We prove that the local height of a point on a Drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module. This gives rise to a Drinfeld module analog of a weak version of Siegel’s integral points theorem over number fields and to an analog of a theorem of Schinzel’s regarding the order of a point modulo certain primes.

An universality theorem on the approximation of analytic functions by shifts ζ(s+iτ, F ) of zeta-functions of normalized Hecke-eigen forms F , where τ takes values from the set {kαh : k = 0, 1, 2, . . . } with fixed 0 < α < 1 and h > 0, is obtained.

L'ubomíra Balková We describe factor frequencies of the generalized Thue–Morse word t b,m defined for b ≥ 2, m ≥ 1, b, m ∈ N, as the fixed point starting in 0 of the morphism ϕ b,m (k) = k(k + 1). .. (k + b − 1), where k ∈ {0, 1,. .. , m − 1} and where the letters are expressed modulo m. We use the result of Frid [4] and the study of generalized Thue–Morse words by Starosta [6].

Introduced by Kraitchik and Lehmer, an x-pseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least x-pseudosquare that improves on a bound that is exponential in x due to Schinzel. We also obtain an equi-distribution result for pseudosquares. An x-pseudopower to base g is a posit...

We study the distribution of the complex sum-of-digits function sq with basis q = −a ± i, a ∈ Z+ for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat [16] for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αsq(p)) provided α ∈ R \Q and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes...

Let P,Q be positive, relatively prime and odd integers such that P 2 − 4Q > 0. We study the sequences (xn)n>0 of positive integers satisfying the recursion formula xn+1 = Pxn − Qxn−1. They generalize the classical Lucas sequences (Un(P,Q)) and (Vn(P,Q)). The prime divisors of Vn(P,Q) for n = 3 · 2 have nice properties which, through the computation of the Legendre Symbols of suitable xn’s modul...