10 - Proof of Goldbach's conjecture


ISSUE No. 10

Vedic Mathematics is becoming increasingly popular as more and more people are introduced to the beautifully unified and easy Vedic methods.
The purpose of this Newsletter is to provide information about developments in education and research and books, articles, courses, talks etc., and also to bring together those working with Vedic Mathematics.
If you are working with Vedic Mathematics- teaching it or doing research- please contact us and let us include you and some description of your work in the Newsletter. Perhaps you would like to submit an article for inclusion in a later issue.
If you are learning Vedic Maths, let us know how you are getting on and what you think of this system.
This issue's article:


We are delighted to include an attachment with this Newsletter which contains a proof, according to Dr S K Kapoor who devised it, of Goldbach's Conjecture: that every even number over two can be expressed as the sum of two primes. The experts in number theory will soon tell us about the validity of Dr Kapoor's proof, which is neat and does not require very advanced mathematics to understand. This is contained in Dr Kapoor's book "Goldbach Theorem" ISBN 81-7063-113-0 and steps are being taken to have it published in a reputable journal.

Christian Goldbach (1690-1764) was born in Prussia and became professor of mathematics and historian of the Imperial Academy at St. Petersburg. He later tutored Tsar Peter II. In his famous letter to the great Swiss mathematician Leonhard Euler dated June 7th 1742 he conjectures that every number that is a sum of two primes can be written as a sum of "as many primes as one wants". Goldbach considered 1 to be a prime number. In the margin of this letter he states his famous conjecture that every number is a sum of three prime numbers and this is equivalent to what is now known as Goldbach's Conjecture: that every even number can be expressed as the sum of two prime numbers. So for example, 30 = 7+23, where 30 is even and 7 and 23 are both prime. As 30 = 11+19 as well, we see that there is more than one way to express this number as a sum of primes.

In fact Dr Kapoor's research suggests that the number of ways of expressing the even number, E, as a sum of primes is equal to at least the square root of E divided by 4. So for E=400 there will be at least 5 ways of partitioning the number 400 into a sum of two primes. The proof follows from this result for E greater than or equal to 64, and the even numbers from 2 to 62 are easily checked.

Dr Kapoor has a first class degree in Mathematics and a Ph.D. in Vedic Mathematics. He was reading The Hindustan Times, Delhi dated March 25, 2000 in which there was an editorial item "From Zero to Infinity" focusing on the challenge of Goldbach's Conjecture. "Straightaway it flashed to me" says Dr Kapoor in the Preface of his book "that the 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". Thus Dr Kapoor used the results of his research in Vedic Mathematics to arrive at the proof. In answer to a question about the use of Vedic Mathematics in the proof" Dr Kapoor replied "As far as your query as to the actual use of VEDIC MATHEMATICS for settlement of the PROOF, the straight answer to it is 'YES'; in fact, it is Vedic Mathematics, or to be specific, Vedic Geometry, and amongst that discipline, the concept of di-monad, a spatial order of manifested layers of dimensional spaces, has supplied the whole format, structural logic and the steps." And he went on to elaborate.

To promote the book "Uncle Petros and Goldbach's Conjecture" by Apostolos Doxiadis, the publishers, Faber and Faber, have put up an award of one million dollars to the first person who can supply a proof of Goldbach's Conjecture within two years. This book revolves around a character, Uncle Petros, who devoted his life to finding a proof of Goldbach's Conjecture and is worth reading. It is interesting to note that Prof. Goldbach himself would not have been able to claim this award if he were alive today and had a proof, as under the rules the award can only go to a resident of the US or the UK! The same applies to Dr Kapoor who lives in India.

The number 2^100 (two to the power of one hundred) is cited in the Uncle Petros book and it is an interesting challenge to show that this number has at least 2^48 ways (remember: square root of E divided by 4) of being partitioned into a sum of two primes. 2^100 is rather a large number however and has rather a lot of prime pairs. But if you can help with this task we or Dr Kapoor () would be very glad to hear from you.

More information about Dr Kapoor and his book can be seen at our web site.





The book "Maharishi's Absolute Theory of Defence" has been added to the books available from the web site at http://www.vedicmaths.com. This book includes a lot of material on Maharishi's Vedic Mathematics. How is mathematics related to defence, you might ask.

"The fundamentals of invincibility can be located everywhere in the field of Mathematics because the nature of mathematical knowledge makes every principle of Mathematics invincible. Nothing can challenge a truth of Mathematics because Mathematics directly validates its knowledge by appeal to pure reason on the basis of the universal dynamics of intelligence, as quantified in Modern Mathematics by the principles of logical inference in conjunction with the fundamental principles of Set Theory, Category Theory, and Topos Theory." (from the above book)

We hope to have an article soon on Maharishi's Vedic Mathematics.


The book "The Cosmic Computer - Abridged Edition" is being reduced in price to 15 pounds (from 20 pounds), for the summer. If you have not got a copy of this excellent introduction to Vedic Mathematics, now is your chance.


The following review entitled "Circles of your mind" was published in "The Teacher" on July/August 2000, page 25.
The Circle Revelation, by A P Nicholas. Published by Inspiration Books. £9.95. ISBN 1902517067.
This book introduces geometry and geometrical reasoning to the casual reader. The thing that differentiates it from other geometry texts is that while most stated results are proved, this is accomplished through dialogue and diagrams that are described simply and are easy to understand.

The foundations of the book lie in Vedic and Indian mathematics which the author has transferred to everyday language. The text in the book is refreshingly hand-written, as are the diagrams.

The first half of the book covers some examples with triangles and circles, building up to rectangles, opposite angles and then chords of circles. A proof of equal angles subtended by a chord is given and then the underlying concepts of the system are discussed.
The second section represents addition and multiplication as diagrams, draws out many angle results from parallelograms, similar triangles and hence parallel lines before branching off into solving linear equations and a nice proof of Pythagoras' Theorem.
Although the dialogue appears to jump from one area of maths to another, these strands often come together in a later result, meaning the book is best read in its entirety.
As a teacher I have already used some of the explanations in lessons. The casual, non-mathematical reader with an interest will be stimulated by this text and its easy to follow explanations, and a student will find the alternative proofs useful.
Nat Parnell, KS3 Co-ordinator, Maths, Plymouth

Author's comment
I am, of course, pleased to have a favourable review from someone who has clearly taken the trouble to understand the book, and also made use of it. The review raises some useful questions in my mind. Is this system based on vedic mathematics? What does that mean? Does it mean 'based on the sutras'? They are not mentioned in 'The Circle Revelation' - although they are discussed at some length in Appendix I of 'Geometry for an Oral Tradition', the more formal book on the topic.
So what is the system based on? Four Provisions (things provided), as discussed in Chapter 4.
They are :-
A language, in use
A means of drawing figures
The ability to recognise valid reasoning
An assumption: that lengths and angles (and areas) are unchanged by movement.
I suggest that the first three of these are requirements of an oral tradition, which originally provided the context for vedic mathematics. It is possible to object that the second Provision is not a requirement of an oral tradition, but sight would have been used freely for learning, wouldn't it? As for the third Provision, reason, is a normal human faculty; we don't need to ask if people have it. The study of Euclidean geometry would be pointless without it, because it is an exercise in reasoning.


The Vedic Maths web site at http://www.vedicmaths.org has been further upgraded thanks to the work of our webmaster Clive Middleton. We now have pop-down menus on the home page and extended tutorials the first of which is interactive. The tutorials will be further developed until they are all interactive.

There are still some minor corrections to be carried out but any feedback would be gratefully received - whether you have suggestions or if the site is not working properly for you. There is a text index if your browser cannot use the features in the pop-down menu version of the site. Please note the old site is still there but it will be removed shortly, so if you have bookmarked pages these will need to be updated.


Barbara Salmon's Creative Maths website at http://www.users.waitrose.com/~dabsalmon/ has also been developed and now has a fascinating "Art Gallery". Worth a visit.


Many thanks to those who offered to help with the translation of the Dutch text referred to in the last Newsletter. This is now being dealt with.


Your comments about this Newsletter are invited.
If you would like to send us details about your work or submit an article for inclusion please let us know on

Articles in previous issues of this Newsletter can be copied from the web site - www.vedicmaths.org:
Issue 1: An Introduction
Issue 2: "So What's so Special about Vedic Mathematics?"
Issue 3: Sri Bharati Krsna Tirthaji: More than a Mathematical Genius
Issue 4: The Vedic Numerical Code
Issue 5: "Mathematics of the Millennium"- Seminar in Singapore
Issue 6: The Sutras of Vedic Mathematics
Issue 7: The Vedic Square
Issue 8: The Nine Point Circle
Issue 9: The Vedic Triangle

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28th July 2000


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