Pythagoras, Fermat and Us

We all know about Pythagoras. But do you know about Fermat? If you don’t know you can read this article. If you know him then you must read this article.

But first, let us go back to 5th standard (I’m sure, given a chance, everyone would like to go back to 5th standard) around the time when we studied Pythagoras theorem (Hope you remember it) and also studied one of its proof. In fact, it is said that, this theorem has been proved in more number of ways than any other theorem (but can you do at least 1 of its proof now?!). Then went over to solve some numerical problems and whoever was able to answer the question: “If hypotenuse of a right angle triangle is 5cm long and sum of other two sides is 7cm then find their length” was considered a genius! Some of us were in that genius shoes but some of us just felt bad and so goes our story.

So what this has got to do with this French guy Fermat? Well he did certainly learn Pythagoras theorem but he didn’t just care about the application of it (Yes application, the only thing which seems to interest us). He saw that there are many triplet of numbers like 3,4,5 or 5,12,13 or many such a,b,c whole numbers satisfying this equation a^2+b^2=c^2. That is we can find many right angle triangles which has whole number units as their lengths. If he had stopped there he wouldn’t have been remembered as Fermat de Pierre. He claimed you cannot find such triplet of whole numbers satisfying a^n+b^n=c^n when n>2. He was somehow able to see why you can’t find such triplets but he just wrote on a margin of a book “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain” and left this world in 1665.

So what? Well the problem was nobody was able to prove him wrong and at the same time nobody knew why he is right. This came to be known as Fermat’s last problem and was in Guinness book as the most difficult problem until 1995 when a British mathematician gave a proof (I won’t tell you the person name who solved it! and you can forget about knowing the proof. Its 100 page long). So this French lawyer who was not even a mathematician questioned the best of mathematicians and mathematically made fun of them.

Where this leaves us? I would say think different and don’t always think of profit from a finding which narrows down our minds. You need not be a mathematician for being a Fermat. All you need is a mind to explore without boundaries. The joy of being able to make fun of the best of minds is priceless.

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4 thoughts on “Pythagoras, Fermat and Us

  1. I wouldn’t wish to dampen your enthusiasm, but it’s really quite untrue to say that Fermat was not a mathematician. He was, yes, professionally a lawyer and he wasn’t paid specifically to do mathematics, but that was the normal state of things for mathematicians in the time he lived: a generic mathematician of the 17th century would be a lawyer or scholar or physician or theologian, often several of these things at once, while simultaneously producing paper after paper (called books in the day) advancing algebra, geometry, the forerunners of calculus, or other fields.

    But before he was thirty, Fermat was restoring ancient mathematical texts so they could be part of the common base of knowledge. By the age of 35 he was embroiled in arguments about the mathematics of falling bodies and the descriptions of spirals with major figures in mathematics, including Galileo; he’d go on to a running dispute with Descartes about optics (and one of his principles — that light travels along the path which requires minimum time to go between two points, is a cornerstone in how the field is now understood). His work on finding tangent lines and maxima and minima were quickly copied and studied, and were leading the way that Newton and Leibniz would follow in forming the calculus as we presently understand it.

    And, yes, he did a good bit of work in what we now term number theory, problems such as identifying classes of prime numbers, or proposing what we now call his Last Theorem (and he was almost certainly mistaken in thinking he had a proof; those better-studied in Fermat’s notebooks than I have reported that a method very similar to ones he later used to show special cases of it looks at first like it would prove the general case, but falls short of actually doing so). But mathematicians are more likely to learn and make use of Fermat’s Little Theorem, because of the use it has in the study of groups.

    1. I appreciate you for giving us a detailed picture of Fermat. I wanted to make the same point. We can be working in various fields no way related to maths (lawyer in Fermat’s case) and just like Fermat we can contribute to pure science or maths without being a paid or full time mathematician or a scientist. We shouldn’t always look for profit or application of whatever we study. I feel maths and science is somewhat divine but in present academia they are projected as money making tools especially here in India.

      However I accept that i probably should have written the some of the lines as “You need not be full time mathematician for being Fermat” and “This French lawyer for whom maths wasn’t his main career”.

      Coming to the debate of if he had a proof. I feel he must have intuitively seen it. Though he may not have written it down. It’s only a personal view.

  2. Kushal i loved reading it….one of the best articles on this blog.

    As I have heard its the most proved theorem in mathematics with more than 380 proofs documented. Every proof gives some new insights of the theorem.

    In fact people used it before Pythagoras proposed the theorem. Even in Vedic scriptures(which date back to much before Pythagoras period) we find the mathematical equations used to construct the ritual ground(which requires to be constructed in exact measures).

    Yeah check out these links:
    http://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples
    which give the general formulas to find the triplets.

    also this which gives almost 100 independent proofs:
    http://www.cut-the-knot.org/pythagoras/index.shtml

    1. Thanks.. I wanted people to explore the links you have given and more details like who solved the problem.. and you can probably find out which is the toughest problem now?

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