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About: Divisor is a(n) research topic. Over the lifetime, 2462 publication(s) have been published within this topic receiving 21394 citation(s). The topic is also known as: factor & submultiple.

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Journal ArticleDOI
Abstract: The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri?Iwaniec?Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri?Iwaniec?Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

512 citations

Journal ArticleDOI
Abstract: We prove that for~${n>30}$, every~$n$-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

332 citations

Journal ArticleDOI
01 Jan 1960
Abstract: In §1 of the present paper, we introduce the notion of a virtual linear system on a non-singular projective surface and we clarify the theories of infinitely near points, of divisors and of linear system with preassigned base conditions. We introduce in §2 the notions of a numerical types and of non-special points with respect to Cremona transformations. They play important roles in §3 in order to prove characterizations and existence theorems of exceptional curves of the first kind and of Cremona transformations. In §4, we introduce the notion of an abnormal curve, and in §5 we give some remarks on superabundance of a complete virtual linear system on a projective plane S. We add some remarks in §6 on the case where the number of base points is at most 9. The recent paper “On rational surfaces, I” in the last volume of our memoirs is quoted as Part I in the present paper. The notations and terminology in Part I are preserved in this paper, except for that the symbol { } for the total transform of a divisor is changed to ( ) ; see §1. We recall here that an $S$ denotes always a projective plane. A curve will mean a positive divisor on a surface. A divisor $c$ on a surface $F$ is identified with a divisor $c'$ on a surface $F'$ if $c=\sum m_{i}c_{i}$ and $c'=\sum m_{i}c'_{i}$ and if $c_{i}$ and $c'_{i}$ are irreducible and are identical with each other as point sets (identification of points is made by natural birational transformations).

288 citations

Book ChapterDOI
28 May 2006
TL;DR: The complexity of recovering the secret key from O(\sqrt p) to O(sqrt d) for Boldyreva's blind signature and the original ElGamal scheme when p–1 has a divisor d ≤p1/2 and d signature or decryption queries are allowed is reduced.
Abstract: Let g be an element of prime order p in an abelian group and $\alpha\in {{\mathbb Z}}_p$. We show that if g, gα, and $g^{\alpha^d}$ are given for a positive divisor d of p–1, we can compute the secret α in $O(\log p \cdot (\sqrt{p/d}+\sqrt d))$ group operations using $O(\max\{\sqrt{p/d},\sqrt d\})$ memory. If $g^{\alpha^i}$ (i=0,1,2,..., d) are provided for a positive divisor d of p+1, α can be computed in $O(\log p \cdot (\sqrt{p/d}+d))$ group operations using $O(\max\{\sqrt{p/d},\sqrt d\})$ memory. This implies that the strong Diffie-Hellman problem and its related problems have computational complexity reduced by $O(\sqrt d)$ from that of the discrete logarithm problem for such primes. Further we apply this algorithm to the schemes based on the Diffie-Hellman problem on an abelian group of prime order p. As a result, we reduce the complexity of recovering the secret key from $O(\sqrt p)$ to $O(\sqrt{p/d})$ for Boldyreva's blind signature and the original ElGamal scheme when p–1 (resp. p+1) has a divisor d ≤p1/2 (resp. d ≤p1/3) and d signature or decryption queries are allowed.

245 citations

Journal ArticleDOI
Abstract: Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any positive integer n there exists an integer r n such that if X is a smooth projective variety of general type and dimension n, then $\phi_{rK_X}\colon X\dasharrow\mathbb{P}(H^0(\mathcal{O}_{X}(rK_X)))$ is birational for all r≥r n .

227 citations

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