Copyright Douglas R. Parker (2016) Copyright 2016 for diagrams and illustrations by Douglas R Parker

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2 Copyright Douglas R. Parker (2016) Copyright 2016 for diagrams and illustrations by Douglas R Parker The right of Douglas R. Parker to be identified as author of this work has been asserted by him in accordance with section 77 and 78 of the Copyright, Designs and Patents Act All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publishers. Any person who commits any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. A CIP catalogue record for this title is available from the British Library. ISBN (Paperback) ISBN (Hardback) ISBN (E-Book) First Published (2016) Austin Macauley Publishers Ltd. 25 Canada Square Canary Wharf London E14 5LQ

3 Acknowledgements This work is dedicated to my family and friends. Many thanks, to all the mathematicians throughout the ages.

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5 Further Books AbSolved, BeSolved, Could it b-solved, DisSolved, ESolved, FreSolved, GenSolved, HeSolved, IsSolved, JosSolved, KaSolved, LozSolved, MeSolved, NuSolved, OhSolved, PreSolved, QSolved, ReSolved, SheSolved, TheeSolved, UnSolved, VesSolved, WyeSolved, XiSolved, YuSolved, Zolved and Zolved Again. Other Titles: Solved Again and Regain Regained Again And Again And Again and Again and A Gain.

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7 Foreword FERMAT S THEOREM A SIMPLE SOLUTION EXPLAINED I have shown two different solutions to this problem in this study, both very simple and both very effective. I have not gone into detail regarding Fermat s history or his last Theorem. I have assumed that the reader would have prior knowledge of Fermat s Last Theorem and may have read the many excellent books already written on the subject. This solution is easy enough for a student with O level maths grade (or below) of algebra to understand. It is explained with the help of young children trying to work out the solution, we shall overhear their conversation throughout this study. The helpers are, in no particular order, Pi Lot who has a very good sense of direction, Wye Watt forever asking questions even for the sake of it, Roland Buhtor, always hungry and needs constant refuelling, Reiss Ply who just likes talking and talking so most of the narrative mostly comes from him, Idi Lot the very young brother of Pi who does not pay attention all the time and plays with his toys and then asks simple questions or gives an insight to their situation, Genny Ius who is the clever

8 one out of them all and lastly, Mathew Matics who only likes logical answers and wants proof of everything. The reader should be aware that there are red herrings and blind passage ways, not all the assumptions are correct, also some of the statements cannot be proved, however, the conclusion does show a simple solution to the problem. It is up to the reader to reach a conclusion; could this be the simple solution that Fermat stated he found? I have only read a few books on Fermat s last theorem, and although they have said that a solution has been found for some numbers they did not provide a proof or how the solution was obtained, mind you, I did not look very far. Also, any resemblance to any person or persons mentioned in this book is only by chance, even you. Enjoy, smile often, laugh as much as you can.

9 BOOK ONE

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11 Contents The Solution An Explanation Results Second Solution Conclusion Epilogue Appendix... 72

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13 The Solution Stage One Wye: So what s this bloke you were talking about the other day? Re: If you are referring to Fermat and his Last Theorem he was a very clever mathematician. Wye: What did he do then? Math: He was very good at maths and left behind a question that has not been answered for over 350 years, he wrote in the margin of one of his papers a riddle that he said he had a very simple solution to. Wye: But nobody s found the answer? Re: Nobody has found HIS answer, the theory has been proven recently by another very clever mathematician who invented a brand new form of maths to prove the answer. Math: I like a good proof, it kind of makes things click together. Wye: So what is this problem? Gen: It states that the equation x n + y n = z n cannot produce a whole number value for z. Id: That s easy, I can do it! 19

14 Pi: No you can t. Id: Yes I can: Pi: No you can t: Id: I think you will find out I can! Pi: We are going off the rails here, let s get back on course, what numbers are we talking about? Gen: We are talking about the smallest values for x and y, which we start at 1, 2, 3 etc. and then we go to infinity, and for the index (n) from 3 to infinity. Wye: Why not start the index at 2 instead of 3? Re: I will explain later, we have not got started on this problem yet. Math: That s a lot of numbers and very big ones, for example a small number might be 12 5 = 248,832. The number 12 is called the base and the number 5 is called the index. A big number might have ten million numeric digits. Id: What is that? Pi: Stop playing and pay attention. That is ten million numbers in a row to form one number. Gen: Yes, it is big but if you multiply that by itself you get an even bigger one. It is a matter of perspective to what you call big, huge vast, etc. There is no limit to infinity. Wye: Where do we start then? Pi: We start with the easy numbers and Pythagoras! Gen: Where s Roland? Re: He s having breakfast. 20

15 For each value of a number (x) there is one corresponding number (y) that will give the equation x 2 + y 2 = z 2, where z is a whole number. All other number values of (y) will produce a decimal value for z. All further progression from x 2 to x 3 and x n etc. will produce a decimal value for ⁿ z n. Therefore the equation x n + y n = z n cannot produce a whole number value for z. Math: These last two statements are not proof, are those a red herring? Id: You get herrings in the sea. Re: Not red ones, you only get read ones in a book! Gen: I thought books were in black and white. Id: You get red herrings from the Red Sea. Pi: Stop this. We need to get back on track, we will have to explain all this later. Math: The first statement is part of Pythagoras s Theory which has been proven, and I believe the second statement to be true. Re: Yes, it will also be explained later. Wye; Really? 21