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This article is cited in 1 scientific paper (total in 1 paper)
Cardinality of subsets of the residue group with nonunit differences of elements
P. V. Roldugin Moscow State Technical University of Radioengineering, Electronics and Automation
Abstract:
The paper is concerned with the subsets $I\subset\left\{ {0,\;\ldots,\;d - 1} \right\}$ for which gcd$\left( {n - m,\;d} \right) \ne 1$ for any $n,\;m \in I$. Such subsets are called sets of nontrivial differences. Let $d > 1$ and ${d_1}$ be the least prime divisor of $d$. We prove that the largest cardinality of a set of nontrivial differences is $d/{d_1}$. 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 $t \leqslant 2$. It is shown that a minimal nonelementary set may have arbitrary order in the interval $\overline {3,\;t} $. The largest cardinality of nonelementary sets is estimated from below and above.
Keywords:
residue group, differences of elements, nonunit elements, subset cardinality.
Received: 17.02.2016
Citation:
P. V. Roldugin, “Cardinality of subsets of the residue group with nonunit differences of elements”, Diskr. Mat., 28:3 (2016), 111–125; Discrete Math. Appl., 27:3 (2017), 187–197
Linking options:
https://www.mathnet.ru/eng/dm1386https://doi.org/10.4213/dm1386 https://www.mathnet.ru/eng/dm/v28/i3/p111
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Abstract page: | 287 | Full-text PDF : | 138 | References: | 38 | First page: | 26 |
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