On sumsets involving $k$th powers of finite fields
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909518180384768 |
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| author | Wu, Hai-Liang Wei, Ning-Liu Li, Yu-Bo |
| author_facet | Wu, Hai-Liang Wei, Ning-Liu Li, Yu-Bo |
| contents | In this paper, we study some topics concerning the additive decompositions of the set $D_k$ of all $k$th power residues modulo a prime $p$. For example, given a positive integer $k\ge2$, we prove that
$$\lim_{x\rightarrow+\infty}\frac{B(x)}{π(x)}=0,$$
where $π(x)$ is the number of primes $p\le x$ and $B(x)$ denotes the cardinality of the set
$$\{p\le x: p\equiv1\pmod k; D_k\ \text{has a non-trivial 2-additive decomposition}\}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_11789 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On sumsets involving $k$th powers of finite fields Wu, Hai-Liang Wei, Ning-Liu Li, Yu-Bo Number Theory In this paper, we study some topics concerning the additive decompositions of the set $D_k$ of all $k$th power residues modulo a prime $p$. For example, given a positive integer $k\ge2$, we prove that $$\lim_{x\rightarrow+\infty}\frac{B(x)}{π(x)}=0,$$ where $π(x)$ is the number of primes $p\le x$ and $B(x)$ denotes the cardinality of the set $$\{p\le x: p\equiv1\pmod k; D_k\ \text{has a non-trivial 2-additive decomposition}\}.$$ |
| title | On sumsets involving $k$th powers of finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2304.11789 |