Cardinality of subsets of the residue group with nonunit differences of elements
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Pavel V. Roldugin
Abstract
The paper is concerned with the subsets I ⊂ {0, ..., d – 1} for which gcd(n – m, d) ≠ 1 for any n, m ∈ I. Such subsets are called sets of nontrivial differences. Let d > 1 and d1 be the least prime divisor of d. We prove that the largest cardinality of a set of nontrivial differences is d/d1. Sets of nontrivial differences in which not all differences of elements are multiples of the same prime factor d are called nonelementary. Let t be the number of prime factors of d. We show that there are no nonelementary sets for t ⩽ 2. It is shown that a minimal nonelementary set may have arbitrary order in the interval
Originally published in Diskretnaya Matematika (2016) 28, №3, 111–125 (in Russian).
References
[1] Prachar K., Primzahlverteilung, Springer-Verlag, Berlin Göttingen Heidelberg, 1957.Search in Google Scholar
[2] Rosser J., Schoenfeld L., "Approximate formulas for some functions of prime numbers”, Ill. J. Math, 6:1 (1962), 64-94.10.1215/ijm/1255631807Search in Google Scholar
[3] Stinson D.R. Muir J.A., "On the low Hamming weight discrete logarithm problem for nonadjacent representations.”, Appl. Alg. in Eng., Commun. and Comput., 16:6 (2006), 461-472..10.1007/s00200-005-0187-7Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Bounds for the average-case complexity of monotone Boolean functions
- Modular algorithm for reducing matrices to the Smith normal form
- Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
- Application of Hadamard product to some combinatorial and probabilistic problems
- Cardinality of subsets of the residue group with nonunit differences of elements
Articles in the same Issue
- Frontmatter
- Bounds for the average-case complexity of monotone Boolean functions
- Modular algorithm for reducing matrices to the Smith normal form
- Distribution of the extreme values of the number of ones in Boolean analogues of the Pascal triangle
- Application of Hadamard product to some combinatorial and probabilistic problems
- Cardinality of subsets of the residue group with nonunit differences of elements
