A generalization of Euler’s Criterion to composite moduli

József Vass
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 3, Pages 9—19
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Authors and affiliations

József Vass
Department of Algebra and Number Theory, Eötvös Loránd University
Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary


A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler’s Criterion through that of Euler’s Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort.


  • Binomial congruences
  • Power residues
  • Generalized primitive roots.

AMS Classification

  • 11A15
  • 11A07
  • 11C08


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Cite this paper

Vass, J. (2016). A generalization of Euler’s Criterion to composite moduli. Notes on Number Theory and Discrete Mathematics, 22(3), 9-19.

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