All Roads Lead To Rome Conjecture

Here I present to you a very simple but interesting conjecture.

Take any natural number n>=5. If n is composite (a number having factors other than 1 and itself), add up all of its prime factors. If n is prime (a number having only two factors: 1 and the number itself), just add one to it. Repeat the process indefinitely. The conjecture states that no matter what number you start with, you shall always eventually reach the 'Perfect Number 6'.

For Example...

(i) Let n=7

     Sequence: {7,8,6}

(ii) Let n=11

     Sequence: {11,12,7,8,6}

(iii) Let n=188

     Sequence: {188,51,20,9,6}

(iv) Let n=1000

     Sequence: {1000,21,10,7,8,6}

(v) Let n=987654

     Sequence: {1799,264,20,9,6}

and so on...

Pythagorean Pairs Theorem

This Theorem states that If a² + b² = c², then the pairs 

^{c+a & b}

^{c-a & b ( c<>a+1)}   

^{c+b & a}

^{c-b & a (c<>b+1)}

<sup>will always have at least one prime factor in common provided a,b,c belong to Integers(Z).

For Example...</sup>

  and so on.....     

Note: '<>' means 'Not Equal To'.