Theorem states that no three positive integers, for example (x,y,z), can satisfy the equation x^n+y^n=z^n if the integer value of n is greater than 2. Fermat’s Last Theorem is an example a Diophantine equation(Weisstein). A Diophantine equation is a polynomial equation in which the solution must be an integers. These equations came from the works of Diophantus who was a mathematician who worked methods on solving these equations. Fermat’s Last Theorem was based on Diophantus’s work.
A more common Diophantine equation would be Pythagorean Theorem, where the solution would be the the Pythagorean triples(Weisstein). However, unlike Pythagorean Theorem, Fermat’s Last Theorem has no practical real world applications. Fermat had scribbled on the margin of Arithmetica, the book that inspired his theorem, that he had a proof that would not fit on the margin of a book. From the 1600’s-mid 1900’s this proof remained unsolved. It was eventually solved by Andrew Wiles. Andrew Wiles as a child always loved math, he would always make up problems and challenge himself.
His greatest challenge was when he stumbled upon Fermat’s Last Theorem at the local library. The problem for Wiles was that on the margin, Fermat did not write the actual proof to the theorem, just that he had a brilliant idea which was too big for a margin to hold. Wiles had to rediscover Fermat’s proof, however he had the information of the other mathematicians who attempted to solve this theorem over the centuries. Ever since Wiles was a teenager, he remained determined to solve this theorem.
When solving the equation seemed impossible, a breakthrough by Ken Ribet linked Fermat’s Last Theorem with The Taniyama-Shimura Conjecture, or the Modular Theorem(Koch). Once he realized this he immediately regained hope, if he was able to solve this then the next step would be Fermat’s Last Theorem. Wiles continuously worked on creating a proof, he worked in complete isolation and told almost nobody about his work in fear of attracting attention. Every day he would think of different methods of creating a proof, it was always on his mind.
For many years, Wiles was completely obsessed with the equation and slowly made progress towards creating a proof. There was no way to tell if his progress would actually help him solve the problem. But eventually in 1993 Wiles solved Fermat’s Last Theorem. Unknown to the public, Wiles had made an error on a major part of the proof. The error was so subtle and abstract, it took a year to revise the proof. The proof is completely different from Fermat’s original proof. The proof Wiles came up with was more than one hundred pages long and required techniques that were not available at the time of Fermat.
Wiles can finally relax now that he solved his childhood obsession, but he still continues to challenge himself with math problems. Finally Andrew Wiles’ mind is at rest(Koch). Pierre De Fermat was born in France, he was a lawyer and an amateur mathematician. Fermat was born into a wealthy family and attended the university of Bordeaux. Fermat was a married man who had five children. Fermat was a busy lawyer and kept math a hobby, never publishing his proofs. His greatest idea, Fermat’s Last Theorem, was not announced by Fermat, but by his son who had stumbled upon his fathers notes in a book.
Fermat is one of the fathers of analytic geometry and probability theory. Fermat contributed in the field of optics, light travel, and calculus(“About,” ). Unfortunately Fermat’s Last Theorem itself has no real world applications, it is simply a theoretical problem. Despite not having any application it helped prove many other ideas that may possibly have some application in society. Fermat’s Last Theorem aided in the advancement in analytic geometry, infinitesimal analysis, number theory, calculus, and probability theory(“Bookrags”).
Unfortunately much of Pierre de Fermat’s work has been lost, and there has also been disputes whether he even had valid proofs. One of the few proofs that helped solve Fermat’s Last Theorem that was written by Fermat was using”infinite descent” to show that the area of a right triangle with integer sides cannot equal the square of an integer. By proving this he was able to show that the equation of Fermat’s Last Theorem was correct when the exponent was equal to four. Therefore eliminating the possibility of four being the exponent value on his equation.
With this information he lowered the amount of numbers that the exponent value could be, moving progress of solving the proof forward. After Fermat was able to prove the special case of four, all he needed to work on was odd primes. To prove that Fermat’s Last Theorem had no solutions when the exponent was greater than two, Fermat proved that for the formula all possible exponents greater than 2 would be a factor of four, would be an odd prime number, or be both. (“Bookrags”) Over the centuries, countless amounts of people attempted to solve Fermat’s Last Theorem.
Many of these people based their research and tactics off of Fermat’s “infinite descent” technique. This technique is a contradiction proof. It is typically used to show that an equation has no solutions. To prove this a person must prove that a set of solutions smaller then the chosen measure is an impossibility. Having this contradiction shows that there are no solutions to the equation(“MathandMultimediaUnderstanding”). The final piece to the century long puzzle that solved Fermat’s Theorem was the connection of Fermat’s Last Theorem to “elliptic curves”.
This idea stated that if Fermat’s Last Theorem had specific points, then the corresponding elliptic curve would have certain properties that would be abnormal, or violated some rules of elliptic curves. By proving this, it would show that the Fermat’s Last Theorem equation would be able to be used to create a “non-modular semi stable elliptic curve”. Wiles proved that all elliptic curves must be modular, and the contradiction was what solved Fermat’s Last Theorem. Pythagorean Theorem and Fermat’s Last Theorem are both examples of Diophantine equations.
Fermat’s Last Theorem is an extension of Pythagorean Theorem,which is x^2+y^2=z^2. As number theory developed, it became known that no Pythagorean triple satisfied this equation if the exponent is greater than 2,which is x^n+y^n=z^n,n>2. Fermat wrote that he had a great proof that was much to lengthy to write down in a margin, but was most likely wrong because modern day technology was required to solve this proof (“MathandMultimediaUnderstanding The Fermat’s Theorem”). Modern technologies are required because the amount of numbers are infinite, and to rove that no number satisfied the equation would take an immense amount of time. Andrew Wiles was not the only person in history who was interested in Fermat’s Last Theorem, countless people have contributed to the solving of Fermat’s Last Theorem. People like Sophie Germain, Ernst Kummer, Louis Mordell, Harry Vandiver, and Samuel Wagstaff also contributed to solving Fermat’s Last Theorem(“Wiles’ Proof on Fermat’s Last Theorem”). Germain was a woman and studied math, which at that time was very uncommon, this is what made her so well known.
She came upon a book of Archimedes and saw that he died trying to solve a problem. She knew then that math must have been a great wonder and immediately began to teach herself. She masqueraded as a man so she could take a public math class. Germain created Germain primes where p is prime and 2xp+1 is primes as well. By having the exponents in Fermat’s Last Theorem equal Germain primes she was able to narrow down the amount of numbers to check for the proof(Koch).
Ernst Kummer first entered the University of Halle wanting to be a minster, he was required to take a math class and found his calling. He won a prize for a magnificent mathematical essay, a teaching career. He taught for many years and made major contributions, he contributed to function and number theory and ideal numbers. His research led to a breakthrough in Fermat’s Last Theorem (Freeman). Louis Mordell proved that Fermat’s equation a a finite amount of integer solutions if the exponent is greater than two.
This lowered the amount of integers that the exponent in Fermat’s equation could be. Mordell was born in America but moved to England. He took a teaching position and created theorems and ideas in number theory(“gap-system”). Harry Vandiver and Samuel Wagstaff were both American born computer scientist Mathematicians. They continued Kummers work with a computer to speed up the process. With the computer, they were able to test all the primes that could fit the exponent value in Fermat’s Last Theorem up to 125,000 and eventually four million(Corry).
Fermat’s Last Theorem was the inspiration for countless amounts of mathematicians over the centuries that it remained unsolved. Fermat’s Last Theorem definetly deserves the title it has earned as one,if not the, most difficult mathematical problems in history. Since its conjuncture centuries ago it interested and baffled mathematicians throughout the world. Although it is unclear if Fermat even knew the proof to his equation, he will still be famed for creating one of the most famous equations in history.
• The Assignment must be submitted on Blackboard (WORD format only) via allocated folder.
• Assignments submitted through email will not be accepted.
• Students are advised to make their work clear and well presented, marks may be reduced for poor presentation. This includes filling your information on the cover page.
• Students must mention question number clearly in their answer.
• Late submission will NOT be accepted.
• Avoid plagiarism, the work should be in your own words, copying from students or other resources without proper referencing will result in ZERO marks. No exceptions.
• All answered must be typed using Times New Roman (size 12, double-spaced) font. No pictures containing text will be accepted and will be considered plagiarism).
• Submissions without this cover page will NOT be accepted.
1. Describe the simple and complex issues pertaining to public management.
Instructions – PLEASE READ THEM CAREFULLY
Submission Date by students: Before the end of Week7
Place of Submission: Students Grade Centre via blackboard.
Weight: 10 Marks
We expect you to answer each question as per instructions in the assignment. You will find it useful to keep the following points in mind. The assignment with be evaluated in terms of your planning, organization and the way you present your assignment. All the three section will carry equal weightage
Kindly read the instruction carefully and prepare your assignment accordingly.
1) Planning: Read the assignments carefully, go through the Units on which they are based. Make some points regarding each question and then rearrange them in a logical order. (3 Marks)
2) Organisation: Be a little selective and analytical before drawing up a rough outline of your answer. Give adequate attention to question’s introduction and conclusion. (3 Marks)
Make sure that:
a) The answer is logical and coherent
b) It has clear connections between sentences and paragraphs
c) The presentation is correct in your own expression and style.
3) Presentation: Once you are satisfied with your answer, you can write down the final version for submission. If you so desire, you may underline the points you wish to emphasize. Make sure that the answer is within the stipulated word limit. (4 Marks)
Write an essay on the following topic in about 1000-1200 words.
“Globalization has changed the functioning of local governments”.
In the light of this statement, discuss the challenges faced by local governments in the age of globalization. With the help of examples, explain how local governmentscan raise their funds, meet future challenges and be able to convert their challenges into opportunities.
(You are required to include at least three scholarly references in your answer).