## Introduction to the Theory of NumbersStarting with the fundamentals of number theory, this text advances to an intermediate level. Author Harold N. Shapiro, Professor Emeritus of Mathematics at New York University's Courant Institute, addresses this treatment toward advanced undergraduates and graduate students. Selected chapters, sections, and exercises are appropriate for undergraduate courses. The first five chapters focus on the basic material of number theory, employing special problems, some of which are of historical interest. Succeeding chapters explore evolutions from the notion of congruence, examine a variety of applications related to counting problems, and develop the roots of number theory. Two "do-it-yourself" chapters offer readers the chance to carry out small-scale mathematical investigations that involve material covered in previous chapters. |

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abelian group arithmetic function arithmetic progressions assertion assume character modulo congruence consider convolution Corollary coset deﬁned Deﬁnition denote derivation direct product Dirichlet divides divisible E 0 mod element equations equivalent Erdos estimate Euler EXERCISES exists ﬁnite abelian group ﬁrst ﬁxed integer following lemma following theorem formula Gauss given integer greatest common divisor hence identity implies induction inﬁnitely integer coefﬁcients inverse Legendre Legendre symbol log log logx logzx lower bound Mobius multiple roots notation number of positive number of solutions O(log O(x log obtain odd perfect number odd prime ord G p a prime polynomial positive integers Prime Number Theorem problem Prove provides quadratic nonresidue Quadratic Reciprocity Law quadratic residue real number reduced residue system relatively prime representation residue classes residues modulo result satisﬁes Show smallest positive solutions modulo squarefree subset sufﬁces sufﬁciently large theory Unique Factorization Theorem yields