I discussed FLT in an earlier post and stated a problem that if solved FLT for will be deduced from there. Below is a solution to that problem. Assume is a “non-trivial” triple of integers that satisfies (1) . By “non-trivial” we mean “". Obviously we can assume that and are positive. If we show for any such triple, there is another triple satisfying the same equation with a smaller , then using the method of “infinite descent” we get a contradiction, and therefore we can deduce that there is no “non-trivial” solutions for . Now one can show that and are relatively prime. Otherwise you can find a “smaller” solution for the equation (1).(more...)
One of the most important achievements of the 20th century in Math is a proof of the Fermat's Last Theorem. It states: Fermat's Last Theorem. Let and be 4 integers such that and . Then . In our previous post we discussed all solutions of this equation for as Pythagorean Triples. Proof of FLT involves advanced Mathematics. Here I would like to bring up a especial case of this theorem for . More generally one can prove the following. Problem. If for 3 integers and , then . One can prove the above statement using infinite descent and combining it with Pythagorean Triples. Think about it if you get a chance. A proof will be posted later.
Any three integers and that can be lengths of sides of a right triangle is a called Pythagorean triple. But we can also think of negative numbers as Pythagorean triples when they satisfy the equation . Here I would like to show you a simple method to find all Pythagorean triples. Asssume is the greatest common divisor of and , i.e. . If we divide and with obviously we get another Pythagorean triples. Therefore to find all Pythagorean triples it is enough to find all triples with g.c.d.=1. Therefore we may assume and have no common factors. One can check that one of or has the same parity as . (why?) Assume and have the(more...)
Infinite descent is a common mathematical method that we use in solving problems and proofs. It is based on a very simple and intuitive fact: every non-empty set of natural numbers has a smallest number. This means if you show a set of natural numbers has no smallest number that set should be empty. This method is commonly used to attack some problems in number theory. Especially for finding solutions to some specific Diophantine equations. I understand that wikipedia is not the most reliable source of information out there, but a lot of their articles are very useful, so I suggest you take a look at this wiki article to find out more about Infinite(more...)