Geodesic
Squares in Piatetski-Shapiro sequences
Kui Liu1 ; Igor E. Shparlinski2 ; Tianping Zhang3
1School of Mathematics and Statistics Qingdao University No. 308, Ningxia Road, Shinan Qingdao, Shandong 266071, P.R. China
2Department of Pure Mathematics University of New South Wales Sydney, NSW 2052, Australia
3School of Mathematics and Information Science Shaanxi Normal University Xi’an, Shaanxi 710119, P.R. China
Acta Arithmetica, Tome 181 (2017) no. 3, pp. 239-252
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We study the distribution of squares in a Piatetski-Shapiro sequence $(\lfloor n^c\rfloor)_{n\in\mathbb N}$ with $c \gt 1$ and $c\not\in\mathbb N$. We also study more general equations $\lfloor n^c\rfloor = sm^2$, $n,m\in \mathbb N$, $1\le n \le N$, for an integer $s$ and obtain several bounds on the number of solutions for a fixed $s$ and on average over $s$ in an interval. These results are based on various techniques chosen depending on the range of the parameters.
DOI : 10.4064/aa8644-8-2017
Keywords: study distribution squares piatetski shapiro sequence lfloor rfloor mathbb mathbb study general equations lfloor rfloor mathbb integer obtain several bounds number solutions fixed average interval these results based various techniques chosen depending range parameters

Kui Liu  1   ; Igor E. Shparlinski  2   ; Tianping Zhang  3

1 School of Mathematics and Statistics Qingdao University No. 308, Ningxia Road, Shinan Qingdao, Shandong 266071, P.R. China
2 Department of Pure Mathematics University of New South Wales Sydney, NSW 2052, Australia
3 School of Mathematics and Information Science Shaanxi Normal University Xi’an, Shaanxi 710119, P.R. China 
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     title = {Squares in {Piatetski-Shapiro} sequences},
     journal = {Acta Arithmetica},
     pages = {239--252},
     year = {2017},
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     number = {3},
     doi = {10.4064/aa8644-8-2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/aa8644-8-2017/}
}
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Kui Liu; Igor E. Shparlinski; Tianping Zhang. Squares in Piatetski-Shapiro sequences. Acta Arithmetica, Tome 181 (2017) no. 3, pp. 239-252. doi: 10.4064/aa8644-8-2017

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