Proceedings of Integers Conference 2009 MODULAR HYPERBOLAS AND THE COEFFICIENTS OF
نویسنده
چکیده
Let Fq be the multiplicative group of a finite field, Fq, of cardinality q, with q odd; and let P(Fq) denote its power set. We define the arithmetical function D : P � Fq � → Z via D(S) = #I � x + x−1, S � −#I � x− x−1, S � , where for S ⊆ Fq , I � x± x−1, S � = � x± x−1 : x ∈ S � . Furthermore, let tq = � k − 1, if q = 4k + 1 k, if q = 4k + 3, and let F (k, l) be the coefficient of xl in (x−1 + 6 + x)k. Then #D−1({l}) = � 2tq(3F (tq, l − 1) + 10F (tq, l) + 3F (tq, l + 1)), q ≡ 1 (mod 4) 2tq(F (tq, l − 1) + 3F (tq, l)), q ≡ 3 (mod 4).
منابع مشابه
ec 2 01 2 COORDINATE SUM AND DIFFERENCE SETS OF d - DIMENSIONAL MODULAR HYPERBOLAS
Many problems in additive number theory, such as Fermat’s last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set A ⊂ Z is considered sum-dominant if |A+A| > |A−A|. If we consider all subsets of {0, 1, . . . , n−1}, as n → ∞ it is natural to expect that almost all subsets should be difference-dominant, as addition is comm...
متن کاملCOORDINATE SUM AND DIFFERENCE SETS OF d-DIMENSIONAL MODULAR HYPERBOLAS
Many problems in additive number theory, such as Fermat’s last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set A ⊂ Z is considered sum-dominant if |A+A| > |A−A|. If we consider all subsets of {0, 1, . . . , n−1}, as n → ∞ it is natural to expect that almost all subsets should be difference-dominant, as addition is comm...
متن کاملConcentration of points on two and three dimensional modular hyperbolas and applications
Let p be a large prime number, K,L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I2(M ; K, L) of solutions of the congruence xy ≡ λ (mod p), K + 1 ≤ x ≤ K + M, L + 1 ≤ y ≤ L + M and for the number I3(M ; L) of solutions of the congruence xyz ≡ λ (mod p), L + 1 ≤ x, y, z ≤ L + M. Using the idea of Heath-Brown from...
متن کاملGeometric Properties of Points on Modular Hyperbolas
Given an integer n 2, let H n be the set H n = {(a, b) : ab ≡ 1 (mod n), 1 a, b n − 1} and let M (n) be the maximal difference of b − a for (a, b) ∈ H n. We prove that for almost all n, n − M (n) = O n 1/2+o(1). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of H n .
متن کاملAn Application of Modular Hyperbolas to Quadratic Residues
For a prime p > 2 let Zp be the group of invertible elements modulo p, and let Hp denote the modular hyperbola xy ≡ 1 (mod p) where x, y ∈ Z. Following [1] we define Hp = Hp ∩ [1, p− 1], that is, Hp = {(x, y) ∈ Z : xy ≡ 1 (mod p), 1 ≤ x, y ≤ p− 1}. We note that the lines l1 : y = x and l2 : y + x = p are lines of symmetry of Hp. In this note we use these two symmetries to prove the following ba...
متن کامل