n o o lim sup z f nn) _ oo, lim inf 1 f n(n) < oa ., (3) n-a w-a f (n) = pa, F(n) = max pin a i Sn p«5n<P-11 (a i , a j)=1 where all the prime factors of the a 's are the prime factors of n. Trivially f (n) S F(n) and f (n) = F(n) if n-= p-but f (n) = F(n) is possible for arbitrarily large co(n), we will give the simple proof later Probably there is no simple characterization of the integers fo...

Apostolico and Ehrenfeucht defined the notion of a maximal quasiperiodic substring and gave an algorithm that finds all maximal quasiperiodic substrings in a string of length n in time O(n log n). In this paper we give an algorithm that finds all maximal quasiperiodic substrings in a string of length n in time O(n log n) and space O(n). Our algorithm uses the suffix tree as the fundamental data...

We consider curves defined over small finite fields with points of large prime order over an extension field. Such curves are often referred to as Koblitz curves and are of considerable cryptographic interest. An interesting question is whether such curves are easy to construct as the target point order grows asymptotically. We show that under certain number theoretic conjecture, if q is a prim...

denote the the maximum gap between consecutive primes less than X . It is clear from the prime number theorem that G(X) > (1 + o(1)) logX, as the average gap between the prime numbers which are 6 X is ∼ logX . Explicit gaps of (at least) this size may be constructed by observing that P (n)+2, P (n)+3, . . . , P (n)+n is a sequence of n−1 composite numbers, where P (n) is the product of the prim...

H"{o}lder in 1893 characterized all groups of order $pqr$ where  $p>q>r$ are prime numbers. In this paper,  by using new presentations of these groups, we compute their full automorphism group.

1. A function is arithmetic if it is defined on the positive integers. Those arithmetic functions which assume real values and satisfy f(ab)-f(a)+f(b) for mutually prime integers a, b are classically called additive. The following examples illustrate the interest of these functions, both for themselves and for their applications. An additive function is defined by its values on the prime powers...

as x → ∞, where L(x) is the function L(x) = ∫ x 1 e t dt. Sarnak established this result by identifying the regulators with lengths of closed geodesics of the modular curve SL2(Z)\H, where H denotes the upper half-plane, and using the Prime Geodesic Theorem for this Riemannian surface. Actually Sarnak proved not this result but the analogue where h(O) is replaced by the class number in the narr...

We consider the problem of determining the maximum and the minimum number of upward planar orientations a maximal planar graph can have. We show that n-vertex maximal planar graphs have at least Ω(n · 1.189) and at most O(n · 4) upward planar orientations. Moreover, there exist n-vertex maximal planar graphs having as few as O(n · 2) upward planar orientations and n-vertex maximal planar graphs...