Has a Simpler Proof of Fermat's Last Theorem Emerged? I Think Yes!

Yes, it looks as though we really have nailed the problem; Fermat’s Last Theorem may very well be not as esoteric and complicated as experience has wanted us to believe. Please forgive us as we may appear to be blowing our own trumpet – we are not persons of those kind of habits, no, we are merely stating facts as they stand insofar as we can best evaluate them. If our proof stands the test of time and experience – as we strongly believe is going to be the case; then, the immortal words of the great physicist Professor Dr. Albert Einstein (1879-1955) that “Subtle is the Lord. Malicious He is not.” may very well resonate with the truth.

Since the theorem was first set into motion in 1637 by the pre-eminent French Lawyer and amateur mathematician – Pierre de Fermat (1607−1665); in a very notorious marginal note in the copy of book Arithmatica, the note which reads:

``It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.”

Fermat’s so-called ``truly marvellous” proof proved costly and elusive to come by. Fermat had the peculiar habit of writing notes in the margins of his books and most of these notes were theorems he himself had discovered and it was left to the readers of these notes to prove the supposed theorems. For some of the theorems, the proofs were discovered amongst his papers. One stood out, and this theorem, even though its proof was not known, it came to be known as Fermat’s Last Theorem.

Because all of the supposed theorems in the margins had had their proofs found amongst Fermat’s papers and in the case were no papers were found, someone had provided the proof, save for the Last Theorem, everyone was of the idea that this last unproved assertion by Fermat had to be true somehow. So, it gained the title Fermat’s Last Theorem.

Failure after failure, even by the three of the very great mathematicians i.e., Italy’s Leonhard Euler (1707-1783), France’s Pierre-Simon, marquis de Laplace (1749 − 1827), and the celebrated genius and Crown Prince of Mathematics i.e., Germany’s Johann Carl Friedrich Gauss (1777 − 1855); this lead to the yet to be proved theorem gaining, “world-wide, international and historic” notoriety as the hardest mathematical problem known to man. Prior to 1995, this problem was recorded and ranked in the Guinness Book of World Records as one of the “most difficult mathematical problems” known to humanity.

It was not until 358 years later that the English Professor working at the University of Cambridge, Sir Andrew Wiles provided the first definitive proof. His proof spans well over 100 pages of paper and it had two flaws before it was accepted as a final proof. This proof by Sir Professor Andrew Wiles employs so advanced mathematics which is only accessible to the esoteric and the highly advanced mathematical minds. We should say, we ourselves do not have the hope of oneday understanding Sir Professor Andrew Wiles proof. We are not mathematicians but physicists and we only delve into the waters of mathematics on a part-time basis only for mental gymnastics.

The proof by Sir Professor Andrew Wiles employs mathematical tools and methods that were only developed after Fermat. This gave rise to the question that if this proof is so hard, “Did Fermat really have the proof as he claimed?” This question has lead to doubts as to whether or not Fermat actually possessed the correct proof of his conjecture. If he really did, then he must have employed mathematical methods and tools that existed during his day.

If anything, our proof does that, it employs mathematical methods and tools that were present in the days of Fermat. Ours – if only its morass substance is what is to be required, it is a proof that spans not more than 5 pages. To add on to that, it employs mathematical methods and tools that were available in the days of Fermat. This leads one to the strong viewpoint that indeed, Fermat might very well have had the proof as he claimed.

As we anxiously await the World to judge our proof, effort and work, we must --- if this be permitted at this point of closing, say that, we are confident that -- simple as it is or may appear, this proof is flawless, it will stand the test of time and experience. It strongly appears that the great physicist and philosopher -- Albeit Einstein, was probably right in saying that ``Subtle is the Lord. Malicious He is not.” If we are proved correct – as we believe we will, then, we dedicate this success to reducing the complexity of the proof to Fermat’s Last Theorem, to the Education system of our beloved motherland -- Zimbabwe.

On a Simpler Proof of Fermat's Last Theorem

In 1995, my heart tore "asunder" when I learnt that Britain's Professor Andrew Wiles had proved Fermat's last theorem. In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers x, y, and z can satisfy the equation x^n + y^n = z^n for any integer value of n greater than two. 

This theorem was first conjectured by great French-man - Pierre de Fermat, in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. He wrote: 

"it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain" 

No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. As a growing child, I kept somewhere at a special corner of my heart that one-day I would discover this proof. 

So, naturally, I was devastated at Professor Andrew Wiles's success because I felt he had taken something that was given to me by God. I was delighted to learn that Professor Andrew Wiles's proof required advanced mathematics that was not available during Fermat's time. So, I told my self there must exist a simpler proof and it is this that I must find and it is this God wants me to find, so as to demonstrated that He is subtler than our minds complicate Him. 

So, now and again, I have had intellectual convulsions sometimes in my sleep where proofs have come and I would jump out of bed to jot them. I would then revise these proofs only to find it is wrong. Today just before dawn, the same happened. I had a convulsion and I immediately jumped out of bed and wrote down the proof. I have gone through it and all appears to be rock solid. I went through the proof in my heard while on the bus to work and all appeared well. I come into my office and went through the proof again, and all appears to be OK. So, I thought of sharing this with you. 

Give me until Monday 9 Sept. 2013, I will tell you if I have nailed it! If so, it will be an achievement for my country!

Why the imbalance between matter and anti-matter in the Universe?

Tuesday 8 July, just before I went into my blankets, a question popped into my heard "Why is the Universe dominated by matter?" Certainly, I am not the first to ask this question, many before me have asked this very question. During the day, this question rang into my heard many times I was kind of disturbed a little. I would look into the the heavens for inspiration and search for an answer. Before going to bed, this same question come again and it felt like it was the first time I was coming across this question. I then looked to my bathroom door and before my eyes, was an equation. I jumped out of the bed, took my red marker and wrote this equation down on the bathroom door. I looked at it and I imagined its properties, I felt I had gotten the solution -- Eureka! I thought of rushing to my office and jotting a paper and quickly send it for publication.

Time had to wait, I went into my blankets but I did not manage to sleep as this equation wonted my thoughts until around about 04h30, I decided to get out of the blankets, take a shower and then rush to the office and jot down the paper. I did this. What is anti-matter? Well, each electronically charged elementary particle has a counterpart with the opposite electronic charge, and this is what is known as the antiparticle. The antiparticle partner of the negative electron, for example, is the positive positron, which was predicted by Paul Dirac in 1930 and discovered by Carl Anderson in 1932.

When a particle and an anti-particle meet, they combine to form a photon and a photon is nothing but light. According to Dirac's Equation which is one of the most beautiful equations one can ever lay their eyes on, there must exist in the Universe the same quantity of matter as anti-matter. If this where the case, you would not be reading this at all because you would be nothing but a bunch of light particles. Thank God, there is no more matter in the Universe than there is matter -- otherwise the Universe would be a bath of light.

So physicist have wondered why the prediction of this beautiful equation of Dirac is not observed. The Russian physicist Andrei Shakolov provided an answer, that if the Universe did not obey mirror symmetry, then, one would in principle, be able to explain this observation of matter/anti-matter imbalance. Can you imagine a Universe that violates mirror symmetry? Well just imagine when you look into the mirror and you see your right hand at the right and the left hand at the left? This would be pretty strange isn't it?!

Well, my musings on the matter reached a conclusion that if one took Dirac's Equation and modified it in such a manner that it incorporated into its structure and framework, a mysteries cosmic energy that fills all of space called the cosmological constant, then one can explain this observation of why we see the dominance of matter over anti-matter. The great physicist and philosopher Albert Einstein was the first to conceive of the cosmological constant. He included it into his equations because he wanted to stop these equations from predicting that the Universe it expanding. As we all know now, he missed a great opportunity to become the founding father of the of the Big Bang Cosmology!

Back to the main theme of this reading -- further, the modification I made to the Dirac Equation, tells us that the very same cosmic energy is response for the instability of anti-matter. Anti-particles appear into our Universe for a brief moment before they disappear. I felt so delighted that I was able to answer this huge cosmic question with a rather simple and trivial modification of this all time beautiful equation.

The paper on this is archived at: http://arXiv.org/abs/0807.1754

Physics, My-life, My-passion

Physics? Why physics? Many people tend to have a negative attitude toward physics saying it is a difficult subject. My attitude to this subject is just the exact opposite -- I guess one must yield to the wise-man's words that one's attitude determines their altitude. I find physics to be so fascinating so much that many at times, I fail to understand why people could have such a negative attitude to this beautiful subject. Sometimes when I look at physics professors, I am tempted to believe the general opinion that physics is a tough subject because the professors who teach it, many at times are completely block-headed vis-a-vis aloof and out of touch with reality -- and most at times, obdurate in the face of new ideas. Because of this, students will naturally find no connection between physics and reality yet everything around us is deeply rooted in physics. Physics is the driving engine to all of technologies and technology is the driving force behind the most industrialized nations and economies of the World.

I can safely say my interest in physics began when I was a little child of eighty. My nickname then was garbage-man (in my local laugage they would call me Madoda-bin) because I would tour all the garbage bins in my area searching them -- for what, I do not know. My tour was redolent of an inspection of The Guard of Honor because I religiously did this everyday until this one beautiful day when all this came to a standstill. I remember I would search these bins, find something, and from these findings I would try to make something beautiful. My parents were people of modest means, my dad a cop [police officer] and my mother a diligent and loving housewife. They hardly bought me toys, just this once at one Christmas when we visited Father Christmas did they [our parents] procure us some toys. Whoever, I did not find these toys to be of interest to me because they lacked what it took to make me happy -- electric components etc. Maybe, because I was this toy-starved little-one, I took it to the garbage bins.

One day my search would come to an end in a telling fashion that made my search look like I was searching for some hidden treasures about the nature of the cosmos and the world. I found in the dustbins amongst a pile of books, three books that would irreversible change my life forever. The first was about the first man that landed on the moon -- Neil Armstrong, with the famous words "It is one step for a man, but a giant leap for mankind". This book had a yellow cover and the pages were a little coarse. The other was a book on African history, on how the Europeans came to Africa and colonised it. This book had a brown cover and the pages were smooth -- the kind of pages that encourages one to read. The other was the Holy Book -- the Bible. It did not have a cover at all and the text was printed in a very small font.

I read the book about Neil Armstrong and I remember that day looking into the sky and then gazing at the moon and saying to my-self, "Has a man landed on the moon?". My imagination was stretched beyond its ordinary limits and it is then that my dream to understand the stars and all there is, was born. I was a quiet child that loved to play on my own and all the time I would try and piece electrical gadgets from the dustbin-finds, in the hope of discovering new devices. The best I did was to make an electric motor and a light bulb and my dad was very supportive of this while my mum was not -- for this very small act alone, my dad holds a special place in heart and my mum holds an unequaled one for having given life to me and having raised the man in me. I remember going to my dad and asking him if I could go to the moon to which he replied "You will have to be a scientist first" and that if I was to achieve this "and you will have go to America.". It was such a thrill. I said to myself, I want to be the first African man to land on the moon but that dream was stolen by Mike Shuttleworth. But, I think its still alive since Shuttle worth went only into space and not to the moon.

When I look back to access my position, I feel I have gone beyond the moon. I have been to the furthest reaches of the Universe just by seating and thinking about the Universe. I learnt of Einstein's lost dream to find a theory that would explain all the forces of nature and ultimately this theory will explain all that is known. I learnt of this in 1995 at high school. I vowed t myself that I would reach for this dream and in 1997 I began my search and to this day the search continues unabated and un-aborted.

The second book, that is the history book, I read it and was taken back in time. When I look back to these times and now that I understand many things about the World and how it operates, I feel this book gave me my well-intended political ambitions to raise up and be part of Africa's political process and as-well, that of the World.

The third book, The Holly Bible, I read from cover to cover. I mean my parents were and are Christians but I had [then] never bothered to read the Bible that they kept in our house -- we attended church religiously and regularly. I read this book and from it I learnt a great deal, to love others the way Jesus Christ loved. This book in great many ways I feel calls me to take up the priesthood or preach the Good news to the people of the World. I don't know if I will do that but one thing is certain, all my life works and deeds will be furnished by love. You will certainly feel a hand of love behind everything I do -- I mean some passion behind all I do and this passion was kindled by the life story and message of Jesus Christ of Nazareth.

These three books gave me the three passions I hold today, the longing for knowledge of the Heavens and the Earth, the longing for love and the pity for the suffering and the longing for a peaceful World where all persons respect and love all other persons. A World where it is understood that true freedom of self and of us all, is when we come to the understanding that it is our sovereign duty to protect, defend and safeguard the divine rights of others -- to celebrate our common humanity embodied in the ecstatic language of love while acknowledging our diversity. The beauty I find and see in nature through physics keeps all my other passions in life alive. I have been able to guard jealously the child-like curiosity and playfulness in me, and this maybe is the reason I find physics such a fascinating subject. I am almost certain that physics is not something for adults and most professors are adults and if they could be like little-children, physics would be fascinating to all.

My Physics Blogspot

Here I will post my physics stuff!