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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the solutions of the congruence $n^{2}\equiv 1~(mod~\phi ^{2}(n))$


Authors: Florian Luca and Michal Krizek
Journal: Proc. Amer. Math. Soc. 129 (2001), 2191-2196
MSC (2000): Primary 11A07, 11A25, 11D09
Published electronically: January 17, 2001
MathSciNet review: 1823899
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Abstract:

In this note, we show that if $n$ is a positive integer satisfying the congruence $n^{2}\equiv 1~ (mod~\phi ^{2}(n))$, then $n\le 3$.


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Additional Information

Florian Luca
Affiliation: Mathematical Institute, Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
Address at time of publication: Instituto de Matemáticas de la UNAM, Campus Morelia, Apartado Postal 61-3 (Xangari), CP. 58 089, Morelia, Michoácan, Mexico
Email: luca@math.cas.cz, fluca@matmor.unam.mx

Michal Krizek
Affiliation: Mathematical Institute, Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
Email: krizek@math.cas.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05929-9
PII: S 0002-9939(01)05929-9
Received by editor(s): November 16, 1999
Published electronically: January 17, 2001
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2001 American Mathematical Society