共查询到20条相似文献,搜索用时 937 毫秒
1.
If r(p) is the least positive integral value of x for which y2 ≡ x(x + 1) ? (x + r ? 1)(modp) has a solution, we conjecture that r(p) ≤ r2 ? r + 1 with equality for infinitely many primes p. A proof is sketched for r = 5. A further generalization to y2 ≡ (x + a1) ? (x + ar) is suggested, where the a's are fixed positive integers. 相似文献
2.
Pierrette Cassou-Noguès 《Journal of Number Theory》1982,14(1):32-64
In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σn?1rP(n)?s ξn, where P ∈ + [X1,…,Xr] and ξn = ξ1n1 … ξrnr, with ξi ∈ , such that |ξi| = 1 and ξi ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over by the ξi and the coefficients of P. If, there exists a number field K, containing the ξi, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a -adic function Z(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k). 相似文献
3.
Christopher Bingham 《Journal of multivariate analysis》1974,4(2):210-223
Define coefficients (κλ) by Cλ(Ip + Z)/Cλ(Ip) = Σk=0l Σ?∈k (?λ) Cκ(Z)/Cκ(Ip), where the Cλ's are zonal polynomials in p by p matrices. It is shown that C?(Z) etr(Z)/k! = Σl=k∞ Σλ∈l (?λ) Cλ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (?λ), ? ∈ k, k ≤ 3. Several identities involving the (?λ)'s are derived. These are used to derive explicit expressions for coefficients of in expansions of P(Z), for all monomials P(Z) in sj = tr Zj of degree k ≤ 5. 相似文献
4.
J.E Nymann 《Journal of Number Theory》1975,7(4):406-412
Given a set S of positive integers let denote the number of k-tuples 〈m1, …, mk〉 for which and (m1, …, mk) = 1. Also let denote the probability that k integers, chosen at random from , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then if k ≥ 3 and where d(S) denotes the natural density of S. From this result it follows immediately that as n → ∞. This result generalizes an earlier result of the author's where and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then as n → ∞. 相似文献
5.
Let = (1, 2, …, n)′ be the least-squares estimator of θ = (θ1, θ2, …, θn)′ by the realization of the process y(t) = Σk = 1nθkfk(t) + ξ(t) on the interval T = [a, b] with f = (f1, f2, …, fn)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and Ds-optimality are used as criteria for choosing f in X. A-, D-, and Ds-optimal models can be constructed explicitly by means of r. 相似文献
6.
Francisco Thaine 《Journal of Number Theory》1985,20(2):128-142
Let p be an odd prime and suppose that for some a, b, c ? \p we have that ap + bp + cp = 0. In Part I a simple new expression and a simple proof of the congruences of Mirimanoff which appeared in his papers of 1910 and 1911 are given. As is known, these congruences have Wieferich and Mirimanoff criteria (2p ? 1 ≡ 1 mod p2 and 3p ? 1 ≡ 1 mod p2) as immediate consequences. Mirimanoff's congruences are expressed in the form of polynomial congruences , 1 ≤ m ≤ p ? 1, and these polynomials Pm(X) are characterized by means of simple relations. In Part II a complement to Kummer-Mirimanoff congruences is given under the hypothesis that p does not divide the second factor of the class number of the p-cyclotomic field. 相似文献
7.
Marvin I. Knopp 《Journal of Number Theory》1980,12(1):2-9
If h, k ∈ Z, k > 0, the Dedekind sum is given by , with , . The Hecke operators Tn for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z+) , where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime. 相似文献
8.
Carl Pomerance 《Journal of Number Theory》1980,12(2):218-223
Let k, l denote positive integers with (k, l) = 1. Denote by p(k, l) the least prime p ≡ l(mod k). Let P(k) be the maximum value of p(k, l) for all l. We show , where γ is Euler's constant and ? is Euler's function. We also show for almost all k. 相似文献
9.
M.M. Robertson 《Journal of Number Theory》1977,9(2):258-270
For 1 ≦ l ≦ j, let l = ?h=1q(l){alh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the alh are positive integers such that j > 1, al1 < … < alq(2) ≦ M, and let l′ = l ∪ {0}. Let p(n : ) be the number of partitions of n = (n1,…,nj) where, for 1 ≦ l ≦ j, the lth component of each part belongs to l and let p1(n : ) be the number of partitions of n into different parts where again the lth component of each part belongs to l. Asymptotic formulas are obtained for p(n : ), p1(n : ) where all but one nl tend to infinity much more rapidly than that nl, and asymptotic formulas are also obtained for p(n : ′), p1(n ; ′), where one nl tends to infinity much more rapidly than every other nl. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the nl tend to infinity at approximately the same rate. 相似文献
10.
Stanley J Benkoski 《Journal of Number Theory》1976,8(2):218-223
If r, k are positive integers, then denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r = 1. An explicit formula for is derived and it is shown that .If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2…psas; pi ∈ S and ai ≥ 0 for all i} and denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for are derived and it is shown that . 相似文献
11.
Let be the Schwartz space of rapidly decreasing real functions. The dual space 1 consists of the tempered distributions and the relation ? L2() ? 1 holds. Let γ be the Gaussian white noise on 1 with the characteristic functional , ξ ∈ , where ∥·∥ is the L2()-norm. Let ν be the Poisson white noise on 1 with the characteristic functional = exp?∫ {[exp(iξ(t)u)] ? 1 ? (1 + u2)?1(iξ(t)u)} dη(u)dt), ξ ∈ , where the Lévy measure is assumed to satisfy the condition ∫u2dη(u) < ∞. It is proved that γ1ν has the same dichotomy property for shifts as the Gaussian white noise, i.e., for any ω ∈ 1, the shift of γ1ν by ω and γ1ν are either equivalent or orthogonal. They are equivalent if and only if when ω ∈ L2() and the Radon-Nikodym derivative is derived. It is also proved that for the Poisson white noice νω is orthogonal to ν for any non-zero ω in 1. 相似文献
12.
Patrick J Browne 《Journal of Differential Equations》1977,23(2):285-292
In this paper we study the linked nonlinear multiparameter system , where xr? [ar, br], yr is subject to Sturm-Liouville boundary conditions, and the continuous functions ars satisfy ¦ . Conditions on the polynomial operators Mr, Prs are produced which guarantee a sequence of eigenfunctions for this problem yn(x) = Πr=1kyrn(xr), n ? 1, which form a basis in . Here [a, b] = [a1, b1 × … × [ak, bk]. 相似文献
13.
For a sequence A = {Ak} of finite subsets of N we introduce: , , where A(m) is the number of subsets Ak ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation constitutes a finite semi-group N∪ (semi-group N∩) (group ). For N∪, N∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr | Akr+1 for r = 1, 2, …. In Section 6 we consider for analogues of Rohrbach inequality: , where g(n) = min k over the subsets {a1 < … < ak} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = ai + aj.Pour une série A = {Ak} de sous-ensembles finis de N on introduit les densités: , où A(m) est le nombre d'ensembles Ak ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations , un semi-groupe fini N∪, N∩ ou un groupe N1 respectivement. Pour N∪, N∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {Akr}, où Akr|Akr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N∪, les analogues de l'inégalité de Rohrbach: , où g(n) = min k pour les ensembles {a1 < … < ak} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = ai + aj. 相似文献
14.
15.
R.J. Cook 《Journal of Number Theory》1979,11(4):505-515
Let where p is a prime ≡ 3 mod 4 and k is an integer ≥ 3. Then S(k) frequently takes large values of each sign. 相似文献
16.
Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number such that there exist different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(pm, ? 2) = (pm ? 2)!, R(pm + 1, ? 3) = (pm ? 2)!, , R(k, ? 0) = k, R(pm, ? 1) = pm(pm ? 1), R(pm + 1, ? 2) = (pm + 1)pm(pm ? 1). The exact value of R(k, ? λ) is determined whenever k ? k0(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k0(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for . 相似文献
17.
Let x?Sn, the symmetric group on n symbols. Let θ? Aut(Sn) and let the automorphim order of x with respect to θ be defined by where xθ is the image of x under θ. Let αg? Aut(Sn) denote conjugation by the element g?Sn. Let where s and k are positive integers and denotes a divides b. Further h(s, k : n) ≡ b(1; s, k : n), where 1 denotes the identity automorphim. If g?Sn let c = f(g, s) denote the number of symbols in g which are in cycles of length not dividing the integer s, and let gs denote the product of all cycles in g whose lengths do not divide s. Then gs moves c symbols. The main results proved are: (1) recursion: if n ? c + 1 and t = n ? c ? 1 then (2) reduction: b(g; s, 1 : c)h(s, 1 : i) = b(g; s, 1 : i + c); (3) distribution: let D(θ, n) ≡ {(k, b) : k?Z+ and b = b(θ; 1, k : n) ≠ 0}; then D(θ, m) = D(φ, m) ∨ m ? N = N(θ, φ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in gss of any given length is smaller than the smallest prime dividing s iff b(gs; s, 1 : c) = 1. If g = (12 … pm)t and then . 相似文献
18.
Rainer Güting 《Journal of Number Theory》1979,11(2):273-278
The wealth of Pythagorean number triples is demonstrated afresh by showing that for every rational number , p, q ∈ there exist infinitely many Pythagorean number triples (a, b, c) which satisfy where a is odd, b is even and a2 + b2 = c2. The special cases where p = 1 or where q = 1 are considered first as they illustrate the method and yield additional results. Rational approximations are also possible by means of the quotients , provided p > q. The results generalize Pythagorean number triples (a, b, c) where a and b differ by a constant investigated by S. Pignataro. 相似文献
19.
Suppose r = (r1, …, rM), rj ? 0, γkj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γk · r = ∑jγkjrj. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑j = 1Naje?λγj · r, where each aj is a nonzero real number. More specifically, if , we study the dependence of . This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when are given. Several examples illustrate the complicated nature of . The results have immediate implication to the theory of stability for difference equations x(t) ? ∑k = 1MAkx(t ? rk) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere. 相似文献
20.
A set {b1,b2,…,bi} ? {1,2,…,N} is said to be a difference intersector set if {a1,a2,…,as} ? {1,2,…,N}, j > ?N imply the solvability of the equation ax ? ay = b′; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b1,b2,…,bi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the equations (, , , (where () denotes the Legendre symbol) and to show that “almost all” sets form both difference and sum intersector sets. 相似文献