We present an intuitively satisfying geometric proof
of Fermat's result for positive integers that for prime
moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in
using the idea of colorings applied to regular polygons to establish a
number-theoretic result. A lemma traditionally, if ambiguously, attributed to
Burnside provides a critical enumeration step.
References
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Gallian, J.A. (2010) Contemporary Abstract Algebra. 7th Edition, Brooks-Cole.
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Rotman, J. (1995) An Introduction to the Theory of Groups. Springer-Verlag. https://doi.org/10.1007/978-1-4612-4176-8
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Burnside, W. (1897) Theory of Groups of Finite Order. Cambridge University Press, Cambridge.
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Frobenius, F.G. (1887) über die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul. Crelle’s Journal, 288.
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Neumann, P.M. (1979) A Lemma That Is Not Burnside’s. The Mathematical Scientist, 4, 133-141.
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Rosyida, I., Peng, J., Chen, L., Widodo, W., Indrati, Ch.R. and Sugeng, K.A. (2016) An Uncertain Chromatic Number of an Uncertain Graph Based on Alpha-Cut Coloring. Fuzzy Optimization and Decision Making, 1-21. https://doi.org/10.1007/s10700-016-9260-x
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Chen, L., Peng, J. and Ralescu, D.A. (2017) Uncertain Vertex Coloring Problem. Soft Computing. https://doi.org/10.1007/s00500-017-2861-7
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