Number Theory Modular Arithmetic Pdf Download [UPDATED]

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This volume contains the proceedings of the Barcelona-Boston-Tokyo Number Theory Seminar, which was held in memory of Fumiyuki Momose, a distinguished number theorist from Chuo University in Tokyo. Momose, who was a student of Yasutaka Ihara, made important contributions to the theory of Galois representations attached to modular forms, rational points on elliptic and modular curves, modularity of some families of Abelian varieties, and applications of arithmetic geometry to cryptography. Papers contained in this volume cover these general themes in addition to discussing Momose's contributions as well as recent work and new results.

In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4.

Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5.

Number theory is an introduction to the study of the number system, primarily the positive integers. It builds the system of numbers from basic axioms and definitions and explores the ramifications and interactions of these basic building blocks. The course is rigorous and proof based.

This course is an introduction to the fundamentals of number theory, with a survey of additional topics that are accessible once the basics are in place. The course is intended to introduce concepts of number theory from a rigorous point of view, and then show students some of the applications of the material they have just learned. Not as deep, but generally broader than a one-semester course for math majors at universities.

The course is lecture oriented with class participation expected. Each major topic begins with a question which leads students to investigate a phenomenon and make conjectures. The class then tries to prove these conjectures, with as much help from the teacher as needed. The theorems often lead to techniques which can be used to solve the initial problem and related problems. The course is two-tiered, in that students can simply learn the ideas and techniques of number theory, or they can delve further into the technical details of the proofs. The course encourages students to improve their ability to write mathematical proofs, but promotes enjoyment of and interest in mathematics as its primary goal. 2b1af7f3a8