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Coprimes or Relatively Prime Numbers :

A pair of numbers whose only common factors ( or greatest common factors ) are 1 or -1. Example -> (15 and 28 ), (9,16)

 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.

Mersenne's Primes :

Prime numbers of the form 2n - 1 where n must be a prime. Example-> 3, 7, 31, 127 , ........ Eucleid proved that if  2n - 1  is a Mersenne's Prime, then 2n-1  is an even, perfect number  and is called Eucleid Number.

Goldbach's Conjecture

The conjecture that every even number (greater than or equal to 6) can be written as the sum of two odd prime numbers.

Prime numbers are the basic building blocks of all numbers.

Fermat's Primes:

We take  every number of the form 2^2^n+1 as the Fermat numbers, and when a number of this form is prime, we call it a Fermat prime

The only known Fermat's Primes are the first 5 Fermat numbers. F(0) = 3, F(1)=5,F(2)=17,F(3)=257 F(4)=65537

F0F1F2.....Fn-1 +2 = Fn.

Among 2 digit numbers from 10 to 99, there are 21 prime numbers--> 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.

Curious Observations:

* 389 is the smallest prime whose sum of digits is an Abundant Number. To understand an abundant number, we take any positive number n and sum its positive divisors. For example, if n=18, sum of its positive divisors S = 1+2+3+6+9+18=39. If S<2n, it is called deficient number. If S>2n, it is called abundant number. If S=2n, it is called perfect number. Here 18 is abundant number.

Infinitude of Primes via Coprime Pairs

For any integer n>1, n and n+1 are coprime - mutually prime, having no common prime factors. So start with any n>1 and write down one of its prime factors, say p. The prime factors of its successor, n+1, are different from p. So there is at least some other prime, say q.

Now consider the successor of the product n(n+1). The prime factors of the latter are different from those of n and n+1, p and q, in particular. Let r be one of those.

Appply the same argument to the successor of n(n+1)[n(n+1)+1] to obtain yet another prime, say s. Obviously the process can be extended indefinitely.