On the sums of squares of exceptional units in residue class rings. (English) Zbl 1538.11026
By using Hensel’s lemma and known results on exponential sums and quadratic Gauss sums, explicit formulas are found for the number of solutions to the congruence \(x_1^2+\cdots+x_k^2\equiv c\) modulo \(n\) with the \(x_i\) being units in the ring \(\mathbb{Z}/n\mathbb{Z}\). The results extend earlier work of M. Mollahajiaghaei [J. Number Theory 170, 35–45 (2017; Zbl 1402.11016)] and Q.-H. Yang and Q.-Q. Zhao [Monatsh. Math. 182, No. 2, 489–493 (2017; Zbl 1425.11018)].
Reviewer: Thomas B. Ward (Durham)
MSC:
| 11B13 | Additive bases, including sumsets |
| 11L03 | Trigonometric and exponential sums (general theory) |
| 11L05 | Gauss and Kloosterman sums; generalizations |
Keywords:
ring of residue classes; quadratic diagonal congruence; exceptional unit; Hensel’s lemma; quadratic Gauss sum; exponential sumReferences:
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