Part 1:

What is Fermats Last Theorem?  Write a small speech (1 - 2 minutes if read out loud) about what Fermats Last Theorem.  Be sure to answer the questions:

 

• What is it?
• Where was it found?
• How old is it?
• Who was Fermat?
• Has it been proved?

 

Part 2:

 

Write a higher level language program that will look for solutions for the equation zⁿ = xⁿ + yⁿ 

for 2 <= n <= 10 and 0 <= x,y <= 100

 

 

 

 

Fermat’s Last Theorem

 

 

My name is Pierre de Fermat and I am being channeled through this student in order to explain the significance of my life and Fermat’s Last Theorem, which has made me famous over the past few centuries.  I was born to a wealthy family in August 20, 1601 in Beaumont-de-Lomagne in France.  My family pushed for me to enter civil service, so I became a lawyer and eventually councilor.  Despite my success in politics I felt a growing interest towards the field of mathematics.  Arithmetica is a compilation of 13 books which contains mathematical problems which have only whole number solutions.  After reading this through, my passion for number theory was solidified.

 

 

We all have our bad habits, and I must admit to taunting fellow mathematicians by asking them to prove theorems, some from texts and some of my own, while stating I have the proofs and refusing to reveal them.  The reason Fermat’s last theorem is named such is because it remained the last of my mathematical conjectures to be proven.  In particular, Pythagoras’ theorem interested me.  The sides of a right triangle can be expressed as x² + y² = z².  One solution to this equation is 3, 4, and 5 – also called a Pythagorean triple.  It has been proven that infinite Pythagorean triples exist.  At around the age of 36, my curiosity led me to make a slight variation to Pythagoras’ theorem.  “Specifically, Fermat wondered whether a cubed version of Pythagoras’ theorem, x³ + y³ = z³, had any solutions.” (Allenbaugh 260)  Thus Fermat’s Last Theorem arose, and the search for “Fermatean triples” began.  Do any solutions exist for x, y, and z such that xⁿ + yⁿ = zⁿ, n > 2?

 

I myself hinted at a solution for n = 4 using the method of infinite descent.  By extension this takes care of multiples such as n = 8, 12, 16, and so on. (Singh 86)  Euler proved n = 3 using i (sqrt(-1)) and infinite descent.  It was later discovered that to prove the conjecture one needed to prove the cases for n = 4 and all prime numbers.  For 350 years after being introduced, Fermat’s Last Theorem is more accurately termed Fermat’s Last Conjecture, as the alleged proof was never revealed.  All that was left behind is the following quote:  “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” (Singh 62)

 

Finally, 357 years later, my theorem was proven.  “Proving the theorem the way it was finally done in the 1990s required a lot more mathematics than Fermat himself could have known.” (Aczel 135)  The proof was finally found by Andrew Wiles of Princeton University and incorporates modular elliptic curves and the Taniyama–Shimura conjecture.  “There are no triples of numbers two adding up to the third where the three numbers are perfect cubes of integers, or fourth powers of integers, fifth, sixth, or any other powers.” (Acze, 43)  The most intriguing aspect of Fermat’s Last Theorem is that it can be understood by a child yet is immensely difficult to prove.  Despite its apparent simplicity many incorrect proofs have been published.

 

 

 

 

 

Works Cited

 

1) Aczel, Amir D.  Fermat’s Last Theorem.  New York:  Four Walls Wight Windows, 1996.

 

 

2) Allenbaugh, Mark H.  “The Enduring and Revolutionary Impact of Pierre de Fermat’s Last Theorem.”  Science and Its Times. Ed.  Neil Schlager and Josh Laurer. Vol 3:  1450 to 1699.  Detroid:  Gale, 2001.  259-262.  Gale Virtual Reference Library.  Gale.  Nassau Community College Library – SUNY.  22 Apr. 2008 http://go.galegroup.com/ps/start.do?p=GVRL&u=sunynassau.

 

3) Singh, Simon.  Fermat’s Enigma.  New York:  Walker Publishing Company, 1997.

 

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