A congruence involving the quotients of Euler and its applications (I)

Cai, Tianxin

doi:10.4064/aa103-4-1

Cited by 15 publications

(19 citation statements)

References 4 publications

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“…In particular, Zudilin pointed me to a few useful ideas and the reference [Gessel and Stanton 1982]. I further thank the anonymous referees for their detailed comments, including pointing out the reference [Cai 2002], on an earlier version of this article.…”

Section: Acknowledgmentsmentioning

confidence: 93%

Hypergeometric evaluation identities and supercongruences

Лонг¹

2011

Pacific J. Math.

135

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Section: Acknowledgmentsmentioning

confidence: 93%

Hypergeometric evaluation identities and supercongruences

Лонг¹

2011

Pacific J. Math.

135

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“…Exact and asymptotic formulas for v n In this appendix we prove the alternative definition (3) and the asymptotic formulae (4) and (10) for the numbers v n defined in (2).…”

Section: Appendix By Don Zagiermentioning

confidence: 96%

“…mod p 2 ). (Indeed, by Morley's congruence (1895), cf [4],. this congruence is even valid modulo p 3 .)…”

mentioning

confidence: 99%

Sequences of Enumerative Geometry: Congruences and Asymptotics

Grünberg¹,

Moree²,

Zagier³

2008

Experimental Mathematics

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We study the integer sequence v n of numbers of lines in hypersurfaces of degree 2n − 3 of P n , n > 1. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the v n are described (in an appendix by Don Zagier). An attempt is made at a similar analysis of two other enumerative sequences: the numbers of rational plane curves and the numbers of instantons in the quintic threefold.We study the sequence of numbers of lines in a hypersurface of degree D = 2n − 3 of P n , n > 1. The sequence is defined by (see e.g.where G(2, n + 1) is the Grassmannian of C 2 subspaces of C n+1 (i.e. projective lines in P n ) of dimension 2(n + 1 − 2) = 2n − 2, Q is the bundle of linear forms on the line (of rank r = 2, corresponding to a particular point of the Grassmannian), and Sym D is its Dth symmetric product -of rank D+r−1 r−1 = D − 1 = 2n − 2. The top Chern class (Euler class) c 2n−2 is the class dual to the 0-chain (i.e. points) corresponding to the zeros of the bundle Sym D (Q), i.e. to the vanishing of a degree D equation in P n ; this is the geometric requirement that the lines lie in a hypersurface.The integral (1) can actually be written as a sum:v n = 0≤i

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“…For positive integer , the generalized Euler function [4] ( ) is defined as the number of positive integers not exceeding [ ] and prime to , namely…”

Section: =1mentioning

confidence: 99%