Abstract
In this chapter, we define and explore the most basic properties of the important relation of congruence modulo n > 1. Our central goal is to prove the famous Fermat’s little theorem, as well as its generalization, due to Euler. The pervasiveness of these two results in elementary Number Theory owes a great deal to the fact that they form the starting point for a systematic study of the behavior of the remainders of powers of a natural number a upon division by a given natural number n > 1, relatively prime with a. We also present the no less famous Chinese remainder theorem, along with some interesting applications.
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Notes
- 1.
A combinatorial proof of Fermat’s little theorem was the object of Problem 5, page 57.
- 2.
After Marie-Sophie Germain, French mathematician of the eighteenth and nineteenth centuries.
- 3.
After John Wilson , English mathematician of the seventeenth century. Although it is a primality test, in practice Wilson’s theorem is not so useful, for, according to Stirling’s formula (cf. [8], for instance), n! grows exponentially as n → +∞.
- 4.
- 5.
If p is prime and \(\alpha \in \mathbb N\), a sufficient condition for the solvability of a congruence of the form f(x) ≡ 0 (mod p α) will be presented in Problem 11, page 394.
- 6.
The hypotheses that the arithmetic progression is nonconstant and has relatively prime initial term and common ratio are actually unnecessary. They were assumed just to simplify the solution of the problem.
References
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
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Caminha Muniz Neto, A. (2018). The Relation of Congruence. In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_10
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DOI: https://doi.org/10.1007/978-3-319-77977-5_10
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