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Erschienen in:
2015 | OriginalPaper | Buchkapitel
7. Numerical Functions and Special Congruences
verfasst von : Sylvia Forman, Agnes M. Rash
Erschienen in: The Whole Truth About Whole Numbers
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Abstract
In this chapter the topic of congruences is expanded to look at three particularly useful theorems on congruences, named after the people who introduced them. The first, Wilson’s Theorem, was originally stated by John Wilson, but the first published proof was by another mathematician, Joseph Lagrange. The second, Fermat’s Little Theorem, was stated by Pierre de Fermat, although it was first proved by Leonhard Euler. The third congruence, known as Euler’s Theorem, was actually stated and proved by Euler himself. Wilson’s Theorem is about factorials. Fermat’s Little Theorem and Euler’s Theorem both provide methods of finding powers of integers that will have a least residue of one in a particular modulus. Both Fermat’s Little Theorem and Wilson’s Theorem require that the modulus is prime, but Euler was able to generalize Fermat’s Little Theorem so that a prime modulus is not required.