共查询到20条相似文献,搜索用时 15 毫秒
1.
Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)|n and pep(n)+1?n. The following results are proved: (1) For any positive integer m, any prime p and any ε∈Zm, there are infinitely many positive integers n such that ; (2) For any positive integer m, there exists a constant D(m) such that if ε,δ∈Zm and p, q are two distinct primes with max{p,q}?D(m), then there exist infinitely many positive integers n such that , . Finally we pose four open problems. 相似文献
2.
Alex V. Kontorovich 《Journal of Number Theory》2006,117(1):1-13
For every positive integer n, the quantum integer [n]q is the polynomial [n]q=1+q+q2+?+qn-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation . 相似文献
3.
Pantelimon St?nic? 《Journal of Number Theory》2003,100(2):203-216
In this paper, we consider the generalized Catalan numbers , which we call s-Catalan numbers. For p prime, we find all positive integers n such that pq divides F(pq,n), and also determine all distinct residues of , q?1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if pq?99999, then is not squarefree for n?τ1(pq) sufficiently large (τ1(pq) computable). Moreover, using the results of the first part, we find n<τ1(pq) (in base p), for which may be squarefree. As consequences, we obtain that is squarefree only for n=1,3,45, and is squarefree only for n=1,4,10. 相似文献
4.
Nikita Sidorov 《Journal of Number Theory》2009,129(4):741-754
Let q∈(1,2); it is known that each x∈[0,1/(q−1)] has an expansion of the form with an∈{0,1}. It was shown in [P. Erd?s, I. Joó, V. Komornik, Characterization of the unique expansions and related problems, Bull. Soc. Math. France 118 (1990) 377-390] that if , then each x∈(0,1/(q−1)) has a continuum of such expansions; however, if , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535-543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m>1 of expansions in base q. In particular, we show that if q<q2=1.71…, then each x has either 1 or infinitely many expansions, i.e., there are no such q in . On the other hand, for each m>1 there exists γm>0 such that for any q∈(2−γm,2), there exists x which has exactly m expansions in base q. 相似文献
5.
Let (|q|<1). For k∈N it is shown that there exist k rational numbers A(k,0),…,A(k,k−1) such that
6.
7.
Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form
8.
Gabriele Ranieri 《Journal of Number Theory》2008,128(6):1576-1586
Let p be an odd prime and q=pm, where m is a positive integer. Let ζq be a qth primitive root of 1 and Oq be the ring of integers in Q(ζq). In [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372-384] I. Gaál and L. Robertson show that if , where is the class number of , then if α∈Oq is a generator of Oq (in other words Z[α]=Oq) either α is equals to a conjugate of an integer translate of ζq or is an odd integer. In this paper we show that we can remove the hypothesis over . In other words we show that if α∈Oq is a generator of Oq then either α is a conjugate of an integer translate of ζq or is an odd integer. 相似文献
9.
Zhi-Wei Sun 《Journal of Number Theory》2007,124(1):57-61
By some extremely simple arguments, we point out the following:
- (i)
- If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n.
- (ii)
- Let OK be the ring of algebraic integers in a quadratic field with d∈{−1,−2,−3,−7,−11}. Then, for any irreducible π∈OK and positive integer k not relatively prime to , there exists a kth power non-residue ω∈OK modulo π such that .
10.
William D. Banks 《Journal of Number Theory》2010,130(3):574-579
For every positive integer n, let be the set of primitive Dirichlet characters modulo n. We show that if the Riemann hypothesis is true, then the inequality holds for all k?1, where nk is the product of the first k primes, γ is the Euler-Mascheroni constant, C2 is the twin prime constant, and φ(n) is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many k for which the same inequality holds and infinitely many k for which it fails to hold. 相似文献
11.
Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a2+b2+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a2+b2=p. 相似文献
12.
Carsten Elsner 《Journal of Number Theory》2007,124(2):442-453
Let sn=1+1/2+?+1/(n−1)−logn. In 1995, the author has found a series transformation of the type with integer coefficients μn,k,τ, from which geometric convergence to Euler's constant γ for τ=O(n) results. In recently published papers T. Rivoal and Kh. & T. Hessami Pilehrood have generalized this result. In this paper we introduce a series transformation with two parameters τ1 and τ2 and integer coefficients μn,k,τ1. By applying the analysis of the ψ-function, we prove a sharp upper bound for |S−γ|. A similar result holds for generalized Stieltjes constants. 相似文献
13.
Sankar Sitaraman 《Journal of Number Theory》2003,99(1):29-35
Let p>5 be a prime number and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1,(k,p)=1.Let Cp(i) be the eigenspace of the p-Sylow subgroup of ideal class group C of corresponding to ωi,ω being the Teichmuller character.In this article we extend the main theorem in Sitaraman (J. Number Theory 80 (2000) 174) and get the following: For any fixed odd positive integer n<p−4, assume:
- (a)
- At least one of Cp(3),Cp(5),…,Cp(n) is non-trivial.
- (b)
- Cp(i)=0 for p−n−1?i?p−2.
- (c)
- for 1?i?n+1.
14.
Non-singular plane algebraic curves over with a Singer group of PGL(3,q) in their automorphism group are classified. Apart from three distinguished points, the set of -rational points of such curves can be partitioned into 2−(q2+q+1,q+1,1) designs each isomorphic to the finite projective plane . 相似文献
15.
Shaun Cooper 《Journal of Number Theory》2003,103(2):135-162
Let rk(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,
16.
John A. Rhodes 《Journal of Number Theory》2003,102(2):278-297
We define n families of Hecke operators for GLn whose generating series are rational functions of the form qk(u)−1 where qk is a polynomial of degree , and whose form is that of the kth exterior product. This work can be viewed as a refinement of work of Andrianov (Math. USSR Sb. 12(3) (1970)), in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially .By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define n families of (polynomial) Hecke operators and characterize their generating series as rational functions. We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting. The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest. We give explicit examples for GL3 and GL4. 相似文献
17.
John Brillhart 《Journal of Number Theory》2004,106(1):79-111
Using the theory of elliptic curves, we show that the class number h(−p) of the field appears in the count of certain factors of the Legendre polynomials , where p is a prime >3 and m has the form (p−e)/k, with k=2,3 or 4 and . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y2+αxy+y=x3 and find an elementary expression for the supersingular polynomial ssp(x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(−p) also shows up in the factorization of certain Jacobi polynomials. 相似文献
18.
Let −D<−4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of exists. Let d be a fundamental discriminant prime to D. Let 2k−1 be an odd natural number prime to the class number of . Let χ be the twist of the (2k−1)th power of a canonical Hecke character of by the Kronecker's symbol . It is proved that the vanishing order of the Hecke L-function L(s,χ) at its central point s=k is determined by its root number when , where the constant implied in the symbol ? depends only on k and ?, and is effective for L-functions with root number −1. 相似文献
19.
Wendt's determinant of order n is the circulant determinant Wn whose (i,j)-th entry is the binomial coefficient , for 1?i,j?n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if p?5, then . Also, if m and n are relatively prime positive integers, then WmWn divides Wmn. 相似文献