Showing papers in "Acta Arithmetica in 1992"
135 citations
116 citations
TL;DR: In this article, it was shown that the affirmative answer to Wall's question implies the first case of FLT (Fermat's last theorem) for exponents which are (odd) Fibonacci primes or Lucas primes.
Abstract: numbers. As applications we obtain a new formula for the Fibonacci quotient Fp−( 5 p )/p and a criterion for the relation p |F(p−1)/4 (if p ≡ 1 (mod 4)), where p 6= 5 is an odd prime. We also prove that the affirmative answer to Wall’s question implies the first case of FLT (Fermat’s last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.
99 citations
72 citations
TL;DR: Theorem 1.1 as mentioned in this paper states that the algebraic function y = y(x) defined by F (x, y) = 0 has only one class of conjugate Puiseux expansions.
Abstract: (C1) ai,n = am,j = 0 for all non-zero i and j, (C2) ai,j = 0 for all pairs (i, j) satisfying ni+mj > mn, (C3) the sum of all monomials ai,jxy of F for which ni+mj = nm is a constant multiple of a power of an irreducible polynomial in Z[x, y]. We note that (C2) is a stronger condition than (C1). The reason that (C1) is included above will be made clear in the statement of Theorem 1. We will make reference to the following condition which, together with (C1), is stronger than (C2) and (C3): (C4) the algebraic function y = y(x) defined by F (x, y) = 0 has only one class of conjugate Puiseux expansions.
61 citations
54 citations
TL;DR: The class number 8 problem was shown to be intractable by Goldfeld, Gross, and Zagier as discussed by the authors, and it was not until the work of Goldfeld and Gross [12, 13] that a general method was developed.
Abstract: We point out that although Baker [2, 3] and Stark [18, 19] succeeded in solving the class number 1 and 2 problems, their methods did not extend to the higher cases. It was not until the work of Goldfeld, Gross, and Zagier [12, 13] that a general method was developed. This new method, however, does not allow one to solve the even class number problems without a good deal of further work. In this regard, we note that Oesterlé [17] finished the class number 3 problem, while the class number 8 problem still appears to be intractable.
50 citations
TL;DR: For a quadratic number x, the above sequence n−1 log qn(x) is always convergent (see Section 2) and its limit is denoted here by β(x), and called the Levy constant of x.
Abstract: One can prove that for a quadratic number x, the above sequence n−1 log qn(x) is always convergent (see Section 2). Its limit is denoted here by β(x) and called the Levy constant of x. We also define the length of x as %(x) = 2 log e0(x) (the terminology will be explained in the course of the paper) where e0(x) = 12 (u0 + √ ∆v0) is the fundamental solution of the Pell equation X −∆Y 2 = 4 . The number ∆ is equal to B−4AC where Ax+Bx+C = 0 is the minimal equation of x in Z, that is, A > 0, A,B,C ∈ Z and (A,B,C) = 1. Our main results in this paper are:
45 citations
43 citations
TL;DR: In this article, the authors considered the problem of the divisor function f(n) = τ(n), and proved that ∆f (x; q, a) = Df(x, q, b, b)−Dm(x; b) x, which yields (1) in the range q < x1//2−2ε.
Abstract: provided x is sufficiently large. An asymptotic formula of type (1) Df (x; q, a) = (1 +O((log x)))Df (x; q) , in which the error term is smaller than the main term by a suitable power of log x, is good enough for basic applications. More important than the size of the error term is the range where (1) holds uniformly with respect to the modulus q. In this paper we consider the problem for the divisor function f(n) = τ(n). In this case one can prove by a simple elementary argument that ∆f (x; q, a) = Df (x; q, a)−Df (x; q) x , which yields (1) in the range q < x1//2−2ε. Using Fourier series technique and Weil’s estimate for Kloosterman sums
42 citations
TL;DR: In this article, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are described.
Abstract: © Foundation Compositio Mathematica, 1992, 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.
TL;DR: The problem of factorisatio numerorum has been studied in the context of additive arithmetical semigroups (see as discussed by the authors for a discussion of the main results and results).
Abstract: Introduction. The problems of “factorisatio numerorum”, which go back more than 65 years, are concerned principally with (i) the total number f(n) of factorizations of a natural number n > 1 into products of natural numbers larger than 1, where the order of the factors is not counted, and (ii) the corresponding total number F (n) of factorizations when the order of the factors is counted. For example, f(12) = 4 while F (12) = 8. Some results and further references on such problems may be found in [1], [2], [5], [6] and [7] in particular. The object of this paper is to consider similar problems and results, with emphasis on the average numbers of factorizations of each kind, within the partly analogous but also quite distinct context of additive arithmetical semigroups. Such semigroups (to be defined below) are treated in the monographs [3], [4] within an abstract setting designed to conveniently cover (under minimal assumptions) concrete cases like (i) the multiplicative semigroup Gq of all monic polynomials in one indeterminate over a finite field Fq, (ii) semigroups of ideals in principal orders within algebraic function fields over Fq, (iii) semigroups formed under direct sum by the isomorphism classes of certain kinds of finite modules or algebras over such principal orders. Although the first main result below will be derived within a still more general framework, we shall formulate and state our second main conclusion within a context (see [4]) which conveniently covers the preceding concrete cases: An (additive) arithmetical semigroup will be understood to be a free commutative semigroup G with identity element 1, generated by a (countable) set P of “prime” elements, which admits an integer-valued “degree”
TL;DR: This paper generalizes a number of the results from [1] by working over Fq where Fq is the finite field of order q and showing that the number of distinct elements generated by a linear recurring sequence is related to the order of its characteristic polynomial.
Abstract: (1) sk = { vk if k ≤ n , ∑n−1 i=0 aisk−n+i if k > n , consists of all nonzero elements of V for k = 1, . . . , p − 1. Such generating patterns are of interest because they provide simple algorithms for generating the linear span of independent subsets of vector spaces over Fp (see [1] for details). In this paper we generalize a number of the results from [1] by working over Fq where Fq is the finite field of order q and by showing that if a0 6= 0, (a0, . . . , an−1) is an n-dimensional generating pattern over Fq if and only if f(x) = x − ∑n−1 i=0 aix i is a primitive polynomial over Fq. More generally, we show that the number of distinct elements generated by a linear recurring sequence is related to the order of its characteristic polynomial. For q = p < 10 with p ≤ 97, we indicate when one can find an optimal n-dimensional generating pattern over Fp with weight two, i.e. with two nonzero ai’s (in [1] the length is defined to be the number of nonzero ai’s but a more natural term is Hamming weight). If V is an n-dimensional vector space over Fq then V is isomorphic to Fqn as a vector space over Fq. Consequently, instead of considering vectors in V as in [1], we may assume that the elements of the sequence are in Fqn . We will make this identification throughout the remainder of the paper.