In the last blog, I spoke about Diophantus's problem: to divide a square into the sum of two smaller squares.
In other words, to find solutions for x,y,z where:
x2 + y2 = z2.
The first step in solving this problem is to realize that we can assume that x,y,z are coprime (or another way to say it, relatively prime). That is, no two of these values are divisible by the same prime. So, if p is a prime that is a factor of x, then we know that it is not a factor of y and not a factor of z.
When we have a situation where the three numbers are not coprime (for example, 6,8,10), we will be able to divide out common factors and end up with three numbers that are.
In the case of 6,8,10, the three numbers share the prime 2. If we divide out 2, then we are left with 3,4,5 which are coprime.
This assumption is important because it greatly simplifies the task of analyzing the conditions for when a solution exists. In my next blog, I will show how this assumption gives us the solution to Diophantus's problem.
Interestingly, we can apply this same assumption to Fermat's Last Theorem. From this point on, we will only need to consider the case where x,y,z are relatively prime.
One of my goals in this project is to provide complete proofs each of the conclusions presented. This blog relies on one lemma. A lemma is an intermediate statement that requires proof and is used in a larger theorem.
Lemma: All solutions to xn + yn = zn can be reduced to a form where x,y,z are coprime. [Here is the proof.]