Institute of Mathematics

of the Czech Academy of Sciences
 
 
 
 

Czechoslovak Mathematical Journal, first online, pp. 1-6

A note on the congruence $x^k\equiv a \pmod n$

R. GOWTHAM, Arindama SINGH, V.P. RAMESH

Received February 5, 2025.   Published online September 17, 2026.
Abstract:  Given a prime $p$, we estimate the $k$th power residues modulo $p^\ell$ from the $k$th power residues modulo $p$. Further, we show that counting the number of $k$th power residues modulo $n$ is one of the elementary consequences of the above estimation.
Keywords:  power residue; congruence; residue system; prime
Classification MSC:  11A05, 11A07, 11A15, 11A41
DOI:  10.21136/CMJ.2026.0032-25
PDF available at:  Springer   Institute of Mathematics CAS
References:
[1] C. F. Gauss: Disquisitiones Arithmeticae. Yale University Press, New Haven (1966). DOI 10.1007/978-1-4939-7560-0 | MR 0197380 | Zbl 0136.32301
[2] T. Nagell: Introduction to Number Theory. John Wiley & Sons, New York (1951). MR 0043111 | Zbl 0042.26702
[3] V. P. Ramesh, R. Gowtham, S. Sinha: A note on cubic residues modulo $n$. Indian J. Pure Appl. Math. 54 (2022), 62-65. DOI 10.1007/s13226-022-00230-z | MR 4550434 | Zbl 1516.11011
[4] S. Seraj: Counting general power residues. Notes Number Theory Disc. Math. 28 (2022), 730-743. DOI 10.7546/nntdm.2022.28.4.730-743
[5] W. Sierpiński: Elementary Theory of Numbers. North-Holland Mathematical Library 31. North-Holland, Amsterdam (1988). MR 0930670 | Zbl 0638.10001
[6] W. D. Stangl: Counting squares in $\Bbb{Z}_n$. Math. Mag. 69 (1994), 285-289. DOI 10.1080/0025570X.1996.11996456 | MR 1424442 | Zbl 1055.11500
Affiliations:   R. Gowtham, Arindama Singh, Department of Mathematics, Indian Institute of Technology Madras, Adyar, Chennai, 600036, Tamil Nadu, India, e-mail: rgowtham.math@gmail.com, asingh@iitm.ac.in V. P. Ramesh (corresponding author), DST-FIST Department of Mathematics, Central University of Tamil Nadu, Neelakudy, Thiruvarur, 610005, Tamil Nadu, India, e-mail: vpramesh@gmail.com
Springer subscribers can access the papers on Springer website.
[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]
© Institute of Mathematics CAS, 2026



 
PDF available at:


Springer
for subscribers
···

Institute of Mathematics CAS
fulltext not available