Many large arithmetic computations rely on tables of all primes less than n. For example, the fastest algorithms for computing n! takes time O(M(n logn) + P(n)), where M(n) is the time to multiply two n-bit numbers, and P(n) is the time to compute a prime table up to n. The fastest algorithm to compute ( n n/2 ) also uses a prime table. We show that it takes time O(M(n) + P(n)). In various mode...

A prime number sieve is an algorithm that nds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O(p n=(log log n) 2) bits of space, an O l (n= log log n) time sieve that uses O(n=((log n) l log log n)) bits of space, where l > 1...

We give two short proofs that for fixed d, a d-regular Cayley graph on an Abelian group of order n has second eigenvalue bounded below by d −O(dn−4/d ), where the implied constant is absolute. We estimate the constant in the O(dn−4/d ) notation. We show that for any fixed d, then for a large odd prime, n, theO(dn−4/d ) cannot be improved; more precisely, most d-regular graphs on prime n vertice...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

We present quantum algorithms for the following graph problems: finding a maximal bipartite matching in time O(n √ m + n log n), finding a maximal non-bipartite matching in time O(n( √ m/n+ log n) log n), and finding a maximal flow in an integer network in time O(min(n √ m · U, √ nUm) log n), where n is the number of vertices, m is the number of edges, and U ≤ n is an upper bound on the capacit...

Let p be a prime and p1, . . . , pr be distinct prime divisors of p− 1. We prove that the smallest positive integer n which is a simultaneous p1, . . . , pr-power nonresidue modulo p satisfies n < p 1/4−cr+o(1) (p → ∞) for some positive cr satisfying cr = e −(1+o(1))r (r → ∞). Mathematical Subject Classification: 11A15, 11A07, 11N29

In the present paper, by considering the notion of ideals in $MV$-algebras, we study some kinds of ideals in $MV$-algebras and obtain some results on them. For example, we present definition of ultra ideal in $MV$-algebras, and we get some results on it. In fact, by definition of ultra ideals, we present new conditions to have prime ideals, positive implicative ideals and maximal ideals in $MV$...

in this paper, various elementary properties of vague rings are obtained. furthermore, the concepts of vague subring, vague ideal, vague prime ideal and vague maximal ideal are introduced, and the validity of some relevant classical results in these settings are investigated.

Abstract In this paper, we study some divisibility properties of palindromic numbers in a fixed base g ≥ 2. In particular, if PL denotes the set of palindromes with precisely L digits, we show that for any sufficiently large value of L there exists a palindrome n ∈ PL with at least (log log n)1+o(1) distinct prime divisors, and there exists a palindrome n ∈ PL with a prime factor of size at lea...