GOLDBACH'S CONJECTURE

The book "GOLDBACH THEOREM" by Dr S. K. Kapoor is published by M/s. Arya Book Depot, 30, Nailwala, Karol Bagh, New Delhi 110005 (INDIA), Phone : (00-91-11)5721221, (00-91-11) 5720363, FAX: (00-91-11)5767012. ISBN-81-7063-113-0. Price : Rs. 175/-.

ABOUT THE BOOK

This book wonderfully simplifies the Proof of 250 years old brain teaser popularly known as Goldbach's conjecture: "Every even number greater than 2 can be written as a sum of two primes." History of the conjecture takes us back to the letter of Christian Goldbach of the year 1742 written by him to Leonhard Euler stating the conjecture. Since then it has remained a brain teaser and unsolved problem of the order of Fermat's last theorem and Riemann Hypothesis. The great mathematicians of the rank and reputation: Hardy, Littlewood and Ramanujan had also been confronted with it but it continued to remain unsolved over all these 250 years.

The great novel "Uncle Petros and Goldbach's conjecture" published by Faber & Faber, UK, earnestly takes us through the absorptions of a mathematician of dedicated spirit like that of Uncle Petros who having devoted whole of his life and till the last breath to capture demonstrable proof of it, though no such demonstrable proof followed.

Now, when the Proof follows, in one of the demonstrations, at Regional Centre, M. D. University, Rewari, the audience, thrilled by the simplicity of the steps of the Proof supplied by Dr. S. K. Kapoor (author of the book) the comments followed conveying:

"When one looks to the very simple steps in which the truth of the conjecture can be demonstrated now even by a non-professional mathematician gives a feeling that somehow everybody missed the proof of this property over 250 years as Nature wanted that credit should go to Dr. Kapoor. The excitement which has been all these years is over as if it should have not been there. And, in any case but for the Proof by Dr. Kapoor it would have remained an excitement to be chased."

ABOUT THE AUTHOR

Dr. S. K. Kapoor is disciple of Yogiraj Sri Sripad Babaji Maharaj, Vrindavan and H. H. Maharishi Mahesh Yogiji Maharaj. Dr. Kapoor is continuing with the old traditions and is reviving the ancient discipline of Vedic mathematics and Vedic Geometry. For his excellence and service rendered in the field of Vedic mathematics, he has been awarded "Shri Guru Gangeshwaranandji Veda Ratna Puraskar-1997" by Bharatiya Vidya Bhavan, Bangalore. The Veda Ratna Puraskar as these are called have come to be accepted as highest epitomes of all honours that a Vedic scholar can strive to achieve in his life time. One may have the idea of the works of Dr. Kapoor by going through Appendix of this book.

Dr. Kapoor was born on February 26, 1946 He graduated as B.Sc. (Hon.) in Mathematics from Kurukshetra University, Kurukshetra in 1965 with distinction of first class first. He post-graduated in Mathematics from Punjab University, Chandigarh in 1967 with distinction. He has been National Scholar throughout. He did his LL.B. in 1970 from Delhi University and joined Haryana Judicial service since 1977. He was awarded Ph.D. in Mathematics from Kurukshetra University, Kurukshetra, on the basis of thesis submitted in 1987. His doctoral thesis is first of its kind and is about the mathematical basis of Vedic literature. He had been Visiting Professor of Indian Institute of Maharishi Vedic Science & Technology from 1988 to 1992.

Presently he is posted as Addl. Distt. & Sessions Judge-I, REWARI 123401 (Haryana).

PREFACE OF THE BOOK

I was already in midst of compilation of my book: "Transcendental basics of Vedic Mathematics of tri-monad format", when the newspaper, The Hindustan Times, Delhi dated March 25, 2k brought the editorial item "From zero to infinity" focusing upon the challenge of Goldbach's conjecture as that "Every even greater than two is sum of a pair of primes". Straight-a-way it flashed to me that format beneath the requirement of the conjecture E = p+q is that of di-monad and by that very evening, the Proof was mentally captured. All these days have been a journey from mental comprehension to the hard printout in the form in which it has reached the present volume.

The present volume is titled as "Goldbach Theorem" as the truth of the conjecture stands established. Those whose anxiety naturally would be first of all to go to the Proof itself, they may straight-a-way go to the last pages of Chapter-1, and then to revert back to the beginning of Chapter-1 to have parallel movement with the evolutionary emergence of the elementary approach culminating into the Proof.

The proof, in a way, has taken us one step forward by providing us the minimum solutions (p,q) for E=p+q; E even, p & q primes as (square root of E)/4 for E>=64. (In this book whenever we refer to minimum solutions (square root of E)/4 we shall be presuming the resriction E>=64.)

My son Rohtash Kapoor has done really a good job by writing suitable computer program and demonstrating the truth of the availability of more than (square root of E)/4 solutions. The result has been tested by the computer program for a reasonably good number of select numbers. The choice has been for small as well as big Even numbers. The biggest Even number tested through computer program is 2^30 = 1,07,37,41,824, a ten digit number. The half of this number is 536870912. By substraction and addition of 5 from it we can have 536870907 and 536870917 whose sum is 1,07,37,41,824. By sequential substraction and addition of 10 from the above pair of numbers, we can reach at the required primes. Like that other Even numbers also can be handled for the required solutions. This aspect has been touched in Chapter-3 titled "Slide rule". Tests of select number with computer program have been given in Chapter-4.

Therefore, in addition to the minimum solutions result ((square root of E)/4), the other result of much interest for the scholars of Number Theory emerging in the process is that E/2 can be handled in terms of its last digit, which always can be suitably attained as 1, 3, 7 or 9 by substraction and addition of small odd numbers in case the last digit is Even. Illustratively, 20=10+10 can be written as 9+11. This property is in fact the property of expression of whole numbers in ten place value system and consequential partitioning of the set of whole numbers into ten subsets with members of the same subsets having common last digit. This in fact, is the property of supplying countably infinite ten flow lines at the origins of ten boundary components (each being hypercube-4) of hypercube-5. As such essentially it is multi-dimensional approach to the number theory on geometric formats of synthetic monads. Chapter-2 of this volume has been specifically devoted to the multi-dimensional approach on synthetic formats to reach at the conceptual proof of the truth of Goldbach's conjecture. This together with the information contained in the Appendix of the book would be of real help to those who are interested to approach the number theory on geometric formats in the Vedic way of higher spaces reality and the Vedic way of the Ganita Sutras. However, Chapter-1, in a way, as is it's title does not presume any results or knowledge beyond the school level understanding and handling of the whole numbers. The approach of Chapter-1 is really elementary and the satisfaction of the author equally lies with the simplicity with which the proof has followed in algorithmic form as well as at the conceptual level on multi-dimensional formats of synthetic monads.

I am highly thankful to my publishers for bringing out such a nice volume within shortest possible time. I am also highly thankful to Rohtash Kapoor who has devoted large number of hours to check the result of minimum number of solutions through computer program. Though he has tested large number of Evens but for want of printing space and to avoid the burden of tabulated information only the select numbers' result have been taken up in Chapter-4. I sincerely feel that the tabulated results of Chapter-4 together with the Slide Rule of Chapter-3 would help the junior as well senior students of Number Theory.

Such a work deserves to be dedicated to the spirit of pursuits of Uncle Petros who devoted his whole life for the truth of the conjecture. I dedicate this work to Uncle Petros' of the past, present and future.

S. K. Kapoor

CONTENTS OF THE BOOK

Chapter One: 1-36

Elementary approach to Goldbach's conjecture

Chapter Two: 37-72

Multi-dimensional approach on di-monad format

Chapter Three: 73-104

Slide rule

Chapter Four: 105-138

Computation of (square root of E)/4 values (p,q) for E=p+q

Appendix 139-184

Transcendental basis of Vedic mathematics

S. K. KAPOOR

Santk@rediffmail.com