Fermat’s Last Theorem

In learning about this subject and thinking about complex concepts, it was pleasant to see the process of discovery. Going beyond academic values it was hart warming to see how some of the most brilliant minds in mathematics struggle with questions.

Fermat was a French judge that enjoyed studying mathematics as a hobby. He was studying Arithmetica which is book of Greek text on mathematics. Habitually, Fermat made notes in the margins of the book about the relevant subject that he was considering. When he arrived at the Pythagorean thermo (a²+b²=c²) and in thinking about the whole numbers that could satisfy the equation (3²+4²=5², 5²+12²=13²…) he advanced to realize that the set of whole numbers that satisfied the solution were infinite. However, almost simultaneously, he seemed to know that we will never find an answer to this equation if we changed the exponents (a³+b³=c³, a⁴+b⁴=c⁴… aⁿ+bⁿ=cⁿ). This is to say so long as the exponent was 2 the answer set is infinite and if the exponent was other than 2 the answer set is empty no matter how far one looks for the answer. But evaluating this idea by trial and error is impossible because when we start with the first number we have infinite number of other numbers to yet check. And after checking the first number we are not one step closer to the end of the set of numbers that we should be examining. This limitless nature of possibilities required a mathematical proof based on other previously established theorems. Surprisingly Fermat notes that he knows such a proof but the margin of the book is too small for the elegant proof that he has imagined.
After Fermat’s passing others look at the notes that he had written in the margins and one by one establish an explanation for each scribble and at last we are left with the problem of variations on the Pythagorean thermo for which no one was able to find a proof. The story remains unresolved until the twentieth century. A young man named Andrew Wiles reads about this in a book from his local library in Cambridge England and tries to solve the problem. He studies math and all through his studies he never forgets about this unsolved problem. Latter in his thirties as a professor he is exposed to a paper by a pair of Japanese mathematicians: Taniyama and Shimura, concerning a conjecture (unproven fact) about a different subject. Professor Wiles invested seven years of complete concentration in proving the Japanese conjecture so that he could use it as a tool to prove Fermat’s theorem.
After Wiles announced that he had found the proof and during the peer review process the complicated procedure was not able to be reproduce and it took another year before the issue could be put to rest.
All Elliptic Curves are Modular => Taniyama and Shimura conjecture => Fermat’s last theorem
So finally, even though professor Wiles proved the Fermat’s theorem, we still do not know what solution Fermat actually had in mind. The techniques that Wiles used were not known during Fermat time.
Works Cited
BBC, Horizons. “Fermat’s Last Theorem”. 11/29/2013 <http://www.youtube.com/watch?v=7FnXgprKgSE>.
Wikipedia Fermat’s Last Theorem Version. 11/29/2013 <http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem>.

Mathematics in Music

Pythagoras, perhaps listening to the sounds of tradesmen at work, started thinking about sounds that were pleasing to hear. We can imagine him watching a blacksmith strike an iron rod to make a sword and noticing that different length blades produced a different sound. Logically he would conclude that the length of a string on an instrument could determine the kind of sound that the instrument produces. These different kinds of sound eventually became known as different notes.
As he went about observing the sound an instrument produced with different length strings, he observed that if the change in length could be represented by a whole number ratio the relating note of music was also pleasing to the ear and harmonious when combined with other notes. Ratios like 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, and 2:1 that today we call Pythagorean Intervals, were at that time recognized for their quality and without the knowledge of sound waves and frequency attributed to gods.
Today we understand wavelengths and can measure their frequencies. If we use a spectrophotometer to measure the frequency of an A note on a violin we should get a reading of 440 hz. Given the 32.5 cm length of the violin string to get to the E note, using Pythagorean Intervals, we shorten the string to 21 and 2/3 cm (32.5*2/3). If we then measure the frequency of the E note, we should get a reading of 660 hz. That “given two notes their frequencies and string length is inversely proportional” is a mathematical representation of the fact that the string producing the E note would be vibrating twice as fast then when it was outputting an A note . Proper fractions and inverse relationships maybe some of the more sophisticated examples of math in music yet at the most basic level each performance starts with “a 1”, “a 2”, “a 1 2 3 4”.

Sports & Mathematics

2 December 2013
From baseball averages to calculating the velocity of a basketball to statistics needed for gambling in sports, math seems to be everywhere. However, math’s graph theory provides a visual scheduling system for teams and players long before any action has taken place on the field. Graph theory classifies a complete graph as one where different points on the graph (vertices) share only one connection. This is to say that if number of points on the graph are represented by n than the number of connections between the points can be predicted to be n-1. If each point is a player or a team, given four players there will be three matches for each player before every player had a chance to face every other player.
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Figure above shows the relationship of player 1 with the other players. Using graph theory, we can show this relationship for all the players with this graph:
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Such a graph maybe an overkill for small number of players but is valuable in more confusing circumstances. The number of connections at each point on the graph is also called the degree of the graph. The degree of the graph above is 3. To calculate the degree of the graph, subtracting one from the number of vertices is easy enough to remember, so I do not know why the formula n[(n-1)/2] is used to complicate the issue! But, this is a good reason to study graph theory in more detail.