Greedy algorithm, arithmetic progressions, subset sums and divisibility
نویسندگان
چکیده
منابع مشابه
Discrepancy of Sums of Three Arithmetic Progressions
The set system of all arithmetic progressions on [n] is known to have a discrepancy of order n1/4. We investigate the discrepancy for the set system S3 n formed by all sums of three arithmetic progressions on [n] and show that the discrepancy of S3 n is bounded below by Ω(n1/2). Thus S3 n is one of the few explicit examples of systems with polynomially many sets and a discrepancy this high.
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Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set [N ] = {1, 2, . . . , N} was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph is the hypergraph of sums of k (k ≥ 1 fixed) arithmetic progressions. The hyperedges of this hypergraph are of the form A1 + A2 + . . . + Ak in [N ], wher...
متن کاملSums of Products of Congruence Classes and of Arithmetic Progressions
Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a+im : i ∈ N0}. For positive integers a, b, c, d,m the sum of products set Rm(a)Rm(b)+Rm(c)Rm(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+lm) for some i, j, k, l ∈ Z}. It is proved that if gcd(a, b, c, d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class R...
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3 Proof of Theorem 1 9 3.1 Estimation of the g1 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Estimation of the g3 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Estimation of the g2 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Putting everything together. . . . . . . . . . . . . . . . . . . . . ....
متن کاملOn Equal Values of Power Sums of Arithmetic Progressions
In this paper, we consider the Diophantine equation b + (a+ b) + · · ·+ (a (x− 1) + b) = = d + (c+ d) + · · ·+ (c (y − 1) + d) , where a, b, c, d, k, l are given integers with gcd(a, b) = gcd(c, d) = 1, k 6= l. We prove that, under some reasonable assumptions, the above equation has only finitely many solutions.
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1999
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(98)00385-9