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Methods of Solving Number Theory Problems (Paperback, Softcover reprint of the original 1st ed. 2018)
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Methods of Solving Number Theory Problems (Paperback, Softcover reprint of the original 1st ed. 2018)
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Through its engaging and unusual problems, this book demonstrates
methods of reasoning necessary for learning number theory. Every
technique is followed by problems (as well as detailed hints and
solutions) that apply theorems immediately, so readers can solve a
variety of abstract problems in a systematic, creative manner. New
solutions often require the ingenious use of earlier mathematical
concepts - not the memorization of formulas and facts. Questions
also often permit experimental numeric validation or visual
interpretation to encourage the combined use of deductive and
intuitive thinking. The first chapter starts with simple topics
like even and odd numbers, divisibility, and prime numbers and
helps the reader to solve quite complex, Olympiad-type problems
right away. It also covers properties of the perfect, amicable, and
figurate numbers and introduces congruence. The next chapter begins
with the Euclidean algorithm, explores the representations of
integer numbers in different bases, and examines continued
fractions, quadratic irrationalities, and the Lagrange Theorem. The
last section of Chapter Two is an exploration of different methods
of proofs. The third chapter is dedicated to solving Diophantine
linear and nonlinear equations and includes different methods of
solving Fermat's (Pell's) equations. It also covers Fermat's
factorization techniques and methods of solving challenging
problems involving exponent and factorials. Chapter Four reviews
the Pythagorean triple and quadruple and emphasizes their
connection with geometry, trigonometry, algebraic geometry, and
stereographic projection. A special case of Waring's problem as a
representation of a number by the sum of the squares or cubes of
other numbers is covered, as well as quadratic residuals, Legendre
and Jacobi symbols, and interesting word problems related to the
properties of numbers. Appendices provide a historic overview of
number theory and its main developments from the ancient cultures
in Greece, Babylon, and Egypt to the modern day. Drawing from cases
collected by an accomplished female mathematician, Methods in
Solving Number Theory Problems is designed as a self-study guide or
supplementary textbook for a one-semester course in introductory
number theory. It can also be used to prepare for mathematical
Olympiads. Elementary algebra, arithmetic and some calculus
knowledge are the only prerequisites. Number theory gives precise
proofs and theorems of an irreproachable rigor and sharpens
analytical thinking, which makes this book perfect for anyone
looking to build their mathematical confidence.
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