Abstract
The exercises in Section 1.2 were concerned with divisibility properties. In many cases rules were studied for a given divisor that gave a new number significantly smaller than the dividend, both having the same remainder upon division by the divisor (e.g., for divisibility by 2, 3, 4, 5, 8, 9, 10, and 11). In general, this relation that two numbers have the same remainder upon division by a given divisor proves to be such a useful tool in number theory that Gauss introduced a special notation to represent it that quickly became part of almost every mathematician’s working vocabulary. Taking the names from Latin, we call two such numbers congruent, and their divisor the modulus.
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References
For the origins of these types of questions, see P. Erdős: Summa Brasiliensis Math. II (1950), pp. 113–123.
P. Erdős, E. Szemerédi: Acta Arithmetica 15 (1968), pp. 85–90.
See the article by P. Erdős referred to in footnote 3, and also S. L. G. Choi: Mathematics of Computation 25 (1971) pp. 885–895.
Lifsic’s proof appears as the solution to Problem 143, p. 43 and pp. 159–161 in N. B. Vasil’ev, A. A. Egorov: Zadachi vsesoyuznikh matematicheskikh Olimpiad (Problems of the All-Soviet-Union Mathematical Olympiads), Nauka, Moscow, 1988 (in Russian).
For a proof of Dirichlet’s theorem we refer the reader to H. Rademacher: Lectures on Elementary Number Theory, Blaisdell Publ. Co., New York 1964. Cf. Chapter 14, pp. 121–136.
Remark of I. Z. Ruzsa, personal communication.
D. Burgess: Mathematika 4 (1957), pp. 106–112.
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Erdős, P., Surányi, J. (2003). Congruences. In: Topics in the Theory of Numbers. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0015-1_2
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DOI: https://doi.org/10.1007/978-1-4613-0015-1_2
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