Triples and quadruples of consecutive squares or non-squares in a finite field
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913977394528256 |
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| author | Cohen, Stephen D. |
| author_facet | Cohen, Stephen D. |
| contents | Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ and similarly for the number of triples of consecutive non-square elements. A key ingredient is the evaluation of Jacobsthal sums over general finite fields by Katre and Rajwade. This extends results of Monzingo(1985) to non-prime fields. Curiously, the same machinery alows the evaluation of the number of consecutive quadruples $\{\al -1, \al,\al+1, \al +2\}$ of square and non-squares over $\F$, when $q$ is a power of 5. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_10737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Triples and quadruples of consecutive squares or non-squares in a finite field Cohen, Stephen D. Number Theory Commutative Algebra 11T30 Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ and similarly for the number of triples of consecutive non-square elements. A key ingredient is the evaluation of Jacobsthal sums over general finite fields by Katre and Rajwade. This extends results of Monzingo(1985) to non-prime fields. Curiously, the same machinery alows the evaluation of the number of consecutive quadruples $\{\al -1, \al,\al+1, \al +2\}$ of square and non-squares over $\F$, when $q$ is a power of 5. |
| title | Triples and quadruples of consecutive squares or non-squares in a finite field |
| topic | Number Theory Commutative Algebra 11T30 |
| url | https://arxiv.org/abs/2402.10737 |