Showing papers on "Divisor published in 1984"

Divisors in residue classes

Abstract: In this paper the following result is proved. Let r, s and n be integers satisfying t) I l/3, gcd(r, s) = 1. Then there exist at most 11 positive divisors of n that arc congruent to r modulo s. Moreover, there exists an efficient algorithm for determining all these divisors. The bound 11 is obtained by means of a combinatorial model related to coding theory. It is not known whether 11 is best possible; in any case it cannot be replaced by 5. Nor is it known whether similar results are true for significantly smaller values of log s/log n. The algorithm trcated in the paper has applications in computational number theory. In this paper we prove the following theorem. THEOREM. Let r, s and n be integers satisfying O n n3, gcd(r,s)= 1. Then there exist at most 11 positive divisors of n that are congruent to r niodulo s, and there is a polynomial algorithm for determining all these divisors. The algorithm referred to in the theorem is described in Section 1. It is polynomial in the sense that the number of bit operations required by the algorithm is bounded by a polynomial function of the binary length of n. More precisely, we shall see that this number of bit operations is O((log n )3). Employing fast multiplication techniques we can improve this bound to O((log n)2?E) for every E > 0. We mention two applications of the algorithm. In several primality testing algorithms (see [3], [7]), the number n to be tested is subjected to a collection of "; pseudo-prime" tests. If n does not pass all these tests it is composite. If n does pass all these tests, one knows that each divisor of n lies in one of a small and explicitly known set of residue classes modulo an auxiliary number s. In the latter case, all divisors of n can easily be found if s satisfies the condition s > n'/2. Our algorithm shows that the same can be done if s satisfies the weaker condition s > nl'/3. In special cases this observation was already made in [2, Theorems 5 and 17]. The second application is to the related problem of factoring n. Choosing s to be a suitable integer exceeding n'/3 and applying our algorithm to all residue classes rmod s, we obtain an algorithm that factors n in time O(n(l/3)+e) for every E > 0. The same bound was achieved by Lehman [6] and, conjecturally, by Pinter [9], by methods that are similar in spirit. There exist better factoring methods, both in theory and in practice (see [7]), but this application indicates at least that it may be difficult to extend the algorithm to significantly smaller values of s. Received April 27, 1983. 1980 Mathematics Subject Classificcation. Primary lOH20, IOA25, 94B25.

Partial Correlation Effects in Direct-Sequence Spread-Spectrum Multiple-Access Communication Systems

Abstract: Nearly all previous analytical results on the performance of direct-sequence spread-spectrum multiple-access (DS/SSMA) communication systems are restricted to systems in which the period p of the signature sequence is equal to the number N of chips per data symbol. In many applications, however, it is necessary to employ signature sequences whose period p is much larger than N . Thus, successive N -chip segments of the signature sequence are used to phase-code the carrier during the transmission of successive data symbols. The performance of such systems depends on the partial correlation properties of the signature sequences (rather than the aperiodic correlation properties as in the case when N=p ). In this paper, we consider the performance of a DS/SSMA system for arbitrary values of p and N . As special cases of our results, we delermine the effects of partial correlation in three classes of systems: those for which N =p , those for which N and p are relatively prime, and those for which N is a divisor of p . We also provide two methods for the design of sequences for systems for which N is a divisor of p .

Method and apparatus for division using interpolation approximation

Abstract: A divide method and a divide apparatus for use in a data processing system. The divisor and dividend are normalized in a normalization circuit. A table unit stores a plurality of approximate reciprocal divisors and differences between adjacent approximate reciprocal divisors and is addressed by the high-order bits of the normalized divisor. The approximate reciprocal divisor read out from the table unit is, in an interpolation approximation circuit, changed into an interpolation approximated approximate reciprocal divisor in accordance with a plurality of bits following the high-order bits and the difference. A multiplication unit multiplies the interpolation approximated approximate reciprocal divisor by the normalized dividend to output a quotient.

Lattice points in a circle and divisors in arithmetic progressions

Abstract: Let A(x) denote the number of lattice points in the circle u2+v2≦x and define θ as the infimum of all reals λ for which\(\bar \bar A(x) = \pi x + 0(x^\lambda )\). The objective of this paper is to show that θ≦35/108 which improves upon all previously known results. This estimate is an immediate consequence of a surprisingly easy generalization of KOLESNIK's work on Dirichlet's divisor problem to divisor functions with respect to arithmetic progressions.