Showing papers on "Divisor published in 1976"

On the number of restricted prime factors of an integer. I

Abstract: We usually write co(n; P)=co(n), ~2(n; P ) = ~ ( n ) . In a previous paper [37], we obtained sharp inequalities for the frequencies of large deviations of co(n; E) and ~(n; E) from their normal order of magnitude. Those inequalities included refinements of a special case of a general theorem due to Elliott [11, Theorem 6] concerning large deviations of/(g(n)), where / is a strongly additive arithmetic function and g(n) is a positive-valued polynomial in n with integral coefficients. Elliott 's result was in turn a refinement (under stronger hypotheses) of a theorem of U~davinis [55]. (The result of U~davinis is stated as Theorem 3.3 in Kubilius [28].) The methods used in [37] were \"almost\" elementary. Here we shall use more difficult methods to obtain asymptotic formulas for large deviations of co(n; E) and ~(n; E). We shall also generalize some of the results of [37] and give some applications. For a partial survey of the literature in this area, see [39]. In order to state our main theorems, it is necessary to introduce further notation which will be used throughout this paper. First, we define

Power free values of polynomials

Abstract: If an integer does not have a k -th power of a positive integer, other than 1, for a divisor, it is said to be k –free. Let f(n) be an irreducible polynomial, with rational integer coefficients, of degree g , having no fixed k -th power divisors other than 1. We define i.e. Nk(x) is the number of positive integers n not exceeding x such that f(n) is k -free. One would expect that f(n) is square-free for infinitely many n and further that, given x sufficiently large, there is an n with x n ≤ x + h , such that f ( n ) is square-free for h = 0 ( x 2 ) where e is any real number > 0. These conjectures, however, seem to be extraordinarily difficult to prove. We begin with a brief account of the best results that have been attained so far.

Non-integer frequency divider having controllable error

Abstract: System and method for dividing an input frequency to produce an output frequency related to the input frequency by a non-integral ratio. The divider ratio is alternated between the integer values bracketing the non-integer ratio according to the accumulated value of known errors associated with each integer divisor value by comparing the accumulated error value to a given value.

On the greatest prime factors of polynomials at integer points

Abstract: © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On some analogues of Titchmarsh divisor problem

Abstract: In [15] Titchmarsh posed and solved under the generalized Riemann Hypothesis, the problem of an asymptotic behavior of the number of the solutions of the equation 1 = p — n 1 n 2 for a prime p ≦ x and natural numbers n 1 and n 2 . When we put then the above problem is to get an asymptotic law for the sum

The class number of the field of 5ⁿth roots of unity

Abstract: Let h~ be the relative class number of the field of 5"th roots of unity. If / is any prime number, then the /-part of h~ is bounded independent of n. Let A: be a number field, K/k the cyclotomic Z^-extension of k, and kn the unique intermediate field of degree p" over k. In [4] it was conjectured that if / ^ p is any prime number then the /-part of the class number of kn is bounded independent of n. This conjecture arose from analogy with the case of function fields over finite fields, where Zp-extensions can be obtained by extending the field of constants. In this case it is not difficult to show that the /-part of the order of the group of divisor classes of degree zero is bounded independent of n [4]. For number fields, the conjecture has been proved when k/Q is abelian and p = 2 or 3. The main obstacle for larger primes p is the existence of p-adic (p — l)st roots of unity, which are, of course, harder to handle when p > 5. In this note we attack the case of the simplest Z5-extension, namely the one obtained by adjoining all 5"th roots of unity, for all n > 1, to the field of 5th roots of unity. Recall that for any imaginary abelian number field K there is a maximal real subfield K +, and since K/K+ is totally ramified (at oo) the class number h+ of K+ divides the class number h of K. The quotient h/h+ is called the relative class number h ~. Theorem. Let h~ be the relative class number of Q(f5n), where f5„ is a primitive 5"th root of unity. Let I be any prime number and let le\\h~. Then e~ is bounded as n -^ oo. In fact, if m, is determined by 5m'||/4 — 1 and n > 2m, + 2, then l\h~/h~_v (la\\b means la\b, la+iJfb). Proof. Since 5 is a regular prime, 5\h~ for any n [2], Therefore, we assume / ¥= 5. The set of odd Dirichlet characters of Q(iV) is obtained as follows: Let Xp X2 be the odd characters for Q(f5)/Q and let xpn be a character of conductor 5" such that xpn(a) depends only on a4 mod 5" (therefore \\>n generates the Received by the editors March 1, 1976. AMS (MOS) subject classifications (1970). Primary 12A50, 12A35. 'Partially supported by NSF Grant MPS74-07491A01. © American Mathematical Society 1977 205 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 206 L. C. WASHINGTON characters of the subfield of Q(£5«) of degree 5"_1 over Q). The odd characters of Q(f5~) are then {X/U^}' where / = 1, 2 and 0 2m, let Tr be the trace function from Fn to Fm. We shall calculate Tr( j 5, ^), where x = Xi or XiNote that Tr(fy) = 0 if c > m. Therefore, Tr(Ua))^0^U"f= l^U"5")= 1 fl4'5" = 1 mod 5" ** a4 = 1 mod 5"'"', in which case Ar(xpn(a)) = 5"~mxpn(a). Therefore,

Factoring Integers Whose Digits Are All Ones

Abstract: Let b be a positive integer greater than one. For each positive integer k, define Ik= b k-I + bk-2 + . + b + 1. Thus Ik is the integer whose base b representation consists of k ones. If (n, b) = 1, i.e., if n and b are relatively prime, we define c(n) to be the smallest positive integer k such that n I Ik (n divides Ik). In our first proposition, we will show that if (n, b) = 1, then c(n) exists and furthermore c (n) ' n. Clearly if n and b are not relatively prime, then n t Ik for any positive integer k. In this paper, we characterize those n for which c(n) = n and those for which c(n) = n 1. We prove that if b A 3 (mod 4), then c(n) = n if and only if each prime factor of n is also a divisor of b 1. In particular, if b = 10, then c(n) = n if and only if n is a power of 3 a fact conjectured by Sobczyk [1]. If b 3 (mod 4), then the condition 4 t n must be added. Finally we prove c (n) = n-i if and only if n is a prime which does not divide b 1, and b is a primitive root of n. If n is a prime, it is easy to determine c(n) (Propositions 2 and 6). Hence, we show (Proposition 5) that it suffices to know c(n) whenever n is a power of a prime. Then we prove (Propositions 7 and 9) that c(pi) j 3, can be determined inductively from the values of c(p) and C(p2). The value of C(p2) is given by Proposition 8. Our main results are stated as Propositions 11 and 12. An interesting question which we leave unanswered is, "For which primes p is I, prime?" That this is indeed a difficult question is clear because when b = 2, this is exactly the problem of determining the Mersenne primes.

Zerlegung total gerichteter Graphen in Kreise

Abstract: The following statements are valid: The complete directed graph ¯Kn, n≡1 (mod 2p), is decomposable into directed 2p-cycles. The complete directed bipartite graph ¯Km,n is decomposable into 2p-cycles if p is a divisor of m and n≧p. If p is a prime, then this condition is necessary, too. The complete directed graph ¯Kn, n≠12, is decomposable into 6-cycles if and only if 6

Averages of Arithmetical Functions

Abstract: The last chapter discussed various identities satisfied by arithmetical functions such as µ(n), ϕ(n), Λ(n), and the divisor functions σ a (n). We now inquire about the behavior of these and other arithmetical functions f(n) for large values of n.

The Number of Unitarily k-Free Divisors of An Integer

Abstract: Let k be a fixed integer ≧ 2. A positive integer n is called unitarily k-free, if the multiplicity of each prime factor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the k-th power of any integer > 1. By a unitary divisor, we mean as usual, a divisor d> 0 of n such that (d, n/d) = 1. The interger 1 is also considered to be unitarily k-free. The concept of a unitarily k-free integer was first introduced by Cohen (1961; §1). Let denote the set of unitarily k-free integers. When k = 2, the set coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1960; §1 and §6). A divisor d > 0 of the positive integer n is called a unitarily k-free divisor of n if d ∈ . Let (n) denote the number of unitarily k-free divisors of n.

A new approach to infinite precision integer arithmetic

Abstract: A new algorithm for integer Greatest Common Divisor calculations has recently been proposed. Although the algorithm can be applied to integers in any base b > 2, it is conjectured to be optimal for b=30, when embedded in a system for symbol manipulation. Representation of the digits in factored form further facilitates the GCD procedure. When choosing the set of residues mod 30 symmetrically with respect to 0, in only 8 out of 29 elements a factor occurs which is different from 2, 3 and 5, the prime divisors of 30. A multiplication and addition table built on the distinction of these two classes of digits will be the intermediary in finding the product in a small number of steps, each involving comparison of 1 or 2 bit quantities. Multiplication in this fashion requires 1/3 of the number of bit manipulations as compared with standard procedures on IBM System/360 and 370, if the latter would be applied to equivalent (i.e. 5-bit) entities. Future implementation of long-integer multiplication is suggested in analogy with an algorithm for multivariate polynomial multiplication. An outline for division on this new basis is included.