Number Theory by Shailesh Shirali echoes from Resonance Selected Articles C S Yogananda Universities Press

135 Pages·2019·1.09 MB·English

by  Shailesh Shirali S Yogananda

Number Theory
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Number Theory
Editors
Shailesh Shirali CS Yogananda
Universities Press (India) Private Limited
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© Universities Press (India) Private Limited 2003
First published 2003
Reprinted 2008, 2010
eISBN 978-81-7371-983-7
e-edition:First Published 2015
ePUB Conversion: Techastra Solutions Pvt. Ltd.
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Contents
Foreword
Preface
1. On Provability versus Consistency in Elementary Mathematics
SHAILESH A SHIRALI
2. To find Four Distinct Positive Integers such that the Sum of Any Two
of them is a Square
S H ARAVIND
3. Bachet's Problem
B BAGCHI
4. Mathematical Induction-An Impresario of the Infinite
BSURY
5. On the Infinitude of Prime Numbers - Euler's Proof
SHAILESH A SHIRALI
6. On Fermat's Two Squares Theorem
SHAILESH A SHIRALI
7. Fermat's Two Squares Theorem Revisited
B BAGCHI
8. Factoring Fermat Numbers -A Unique Computational Experiment for
Factoring F
9
C E VENIMADHAVAN
9. The Class Number Problem - Binary Quadratic Forms
RAJAT TANDON
10. The Class Number Problem - An Introduction to Algebraic Number
Theory
RAJAT TANDON
11. Roots are Not Contained in Cyclotomic Fields
RAJAT TANDON
12. Die ganzen Zahlen hat Gott gemacht - Polynomials with Integer
Values
B SURY
13. Prime Representing Quadratics
N V TEJASWI
14. The Congruent Number Problem
V CHANDRASEKAR
15. Fermat's Last Theorem-A Theorem at Last!
C S YOGANANDA
16. Some Unsolved Problems in Number Theory - Progress Made in
Recent Times
K RAMACHANDRA
When and Where the Articles Appeared in Resonance
Foreword
The Indian Academy of Sciences launched Resonance as a monthly journal
devoted to science education in January 1996. Resonance is aimed largely at
undergraduate students and teachers of science, though material of interest to
somewhat younger students is also included. Each issue contains papers that
span a wide area of science and mathematics, in various formats. Some are
individual general articles, others consist of series with several parts. An effort
is made to ensure good expository quality in all of them.
"Echoes from Resonance" is a series of books born out of Resonance, by
putting together in a coherent manner a collection of articles (both series and
single pieces) taken from Resonance, all written around a common theme.
Typically, the individual articles would have appeared quite independently at
different times. These collections should prove useful to a reader who is keen
to learn about a specific subject, with accounts given by different authors
from different perspectives, but all in an expository manner. We hope these
volumes would be useful for students and teachers alike, and that they will
complement the structure of individual issues of Resonance which cover
different areas of science and mathematics in a balanced manner.
N. MUKUNDA
Preface
Number theory has been a subject of study by mathematicians from the most
ancient of times. In the Plimpton 322 clay artefact, excavated from the ruins
of ancient Babylon, one finds a systematic listing of a large number of
Pythagorean triples—triples (a, b, c) of positive integers such that a2 + b2 =
c2; they appear to be listed in order of increasing c∕a ratio. (One sees in the
table the beginnings of trigonometry.) The Greeks had a deep interest in
number theory. Euclid’s great text, The Elements, generally considered as a
book only on Geometry, actually contains a fair amount of number theory
too; in particular it contains the proofs of two gems discovered by the
Greeks–the irrationality of 2 and the infinitude of the primes. It also contains
a description of the algorithm now known as the Euclidean algorithm, which
computes the greatest common divisor of two given numbers. In ancient
India too there was much interest in number theory, particularly in
Diophantine equations; for instance, in the linear two-variable equation ax +
by = c, where a,b,c are given integers, and in the equation later to be known as
the Pell equation (x2 – Ny2 = 1, where N is a given positive integer). Building
on the work of Brahmagupta (6th century) Bhaskara II (12th century) gave a
completely general way of solving the latter equation.
In this book we offer the reader some articles in number theory that
appeared in Resonance over the years 1996–2001. Traditionally, number
theory begins with a study of congruences (Wilson’s and Fermat’s theorems,
the Chinese remainder theorem, quadratic residues, primitive roots, …), then
proceeds to a study of prime numbers (the infinitude of various classes of
primes, divergence of the sum ∑ 1∕p taken over all primes, …) and later to a
study of Diophantine equations (solution of equations such as x2 + y2 = z2,
ax + by = c, where a,b,c are given integers, Pell’s equation …). The last two
topics (prime numbers, Diophantine equations) are distinguished by the
extraordinary diversity in terms of level of difficulty, of the problems they
offer to the students. There is something in number theory for practically