Table Of Content

Echoes from R e^s c & Selected Articles from Resonance—A Journal of Science Education Editors Shailesh Shirali + C S Yogananda INDIAN ACADEMY OF SCIENCES Digitized by the Internet Archive in 2018 ) https://archive.org/details/numbertheoryOOOOunse Number Theory Number Theory Editors Shailesh Shirali + C S Yogananda INDIAN ACADEMY OF SCIENCES Universities Press Universities Press (India) Private Limited Registered Office 3-5-819 Hyderguda, Hyderabad 500 029 (A.P.), India Distributed by Orient Longman Private Limited Registered Office 3-6-752 Himayatnagar, Hyderabad 500 029 (A.P.), India Other Offices Bangalore / Bhopal / Bhubaneshwar / Chennai Emakulam / Guwahati / Hyderabad / Jaipur / Kolkata Lucknow / Mumbai / New Delhi / Patna © Universities Press (India) Private Limited 2003 ISBN 81 7371 454 1 Typeset by Devi Information Systems Chennai Printed at Orion Printers Hydreabad 500 004 Published by Universities Press (India) Private Limited 3-5-819 Hyderguda, Hyderabad 500 029 (A.P.)> India Contents Foreword. vii Preface. ix 1. On Provability versus Consistency in Elementary Mathematics. 1 SHAILESH A Shirali 2. To find Four Distinct Positive Integers such that the Sum of Any Two of them is a Square. 5 S H ARAVIND 3. Bachet’s Problem. 7 B Bagchi 4. Mathematical Induction—An Impresario of the Infinite. 10 B SURY 5. On the Infinitude of Prime Numbers — Euler's Proof. 16 Shailesh A Shirali 6. On Fermat’s Two Squares Theorem. 30 Shailesh A Shirali 7. Fermat’s Two Squares Theorem Revisited. 34 B Bagchi 8. Factoring Fermat Numbers — A Unique Computational Experiment for Factoring Fg. 41 C E Veni Madhavan 9. The Class Number Problem — Binary Quadratic Forms. 43 Raj at Tandon 10. The Class Number Problem — An Introduction to Algebraic Number Theory . 48 Rajat Tandon 11. Roots are Not Contained in Cyclotomic Fields. 56 Rajat Tandon 12. Die ganzen Zahlen hat Gott gemacht — Polynomials with Integer Values. 60 B SURY v vi Number Theory 13. Prime Representing Quadratics. 71 N V TEJASWI 14. The Congruent Number Problem. 73 V CHANDRASEKAR 15. Fermat's Last Theorem —A Theorem at Last!. 83 C S Yogananda 16. Some Unsolved Problems in Number Theory — Progress Made in Recent Times. 91 K Ramachandra When and Where the Articles Appeared in Resonance. 97 Index. 99 Foreword The Indian Academy of Sciences launched Resonance as a monthly journal devoted to science education in January 1996. Resonance is aimed largely at under¬ graduate 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 arti¬ cles, 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 bom 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 collec¬ tions 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