Explanation for the Number Cycle that was Introduced by Mohammadreza Barghi

Mohammadreza Barghi

Abstract: Previously I introduced a number cycle that takes two consecutive integers and a third integer that after a cycle of some operations including adding and dividing, returns one of the two consecutive integers. Here I am explaining how it works. If you add integer "a" to integer "b" and then divide it by two, the result could be written as follows i.e.: (b + a) / 2 = ( b + a) /2 +( - a + a) = (b/2 + a/2 - a) + a = ( (b-a)/2 + a. In the next operations, this result substitutes "b", and if you add this new "b" to "a" and divide it by 2, it will be: (((b - a)/2 + a) + a)/2⤏((b -a)/2 + a + a)/2⤏(b - a)/4 + ( a + a)/2⤏(b - a) / 4 + a. Repeating this cycle causes that (b - a)/2^n (n is the number of repeating of (b -a)/2) ends to 1 or -1 depending if b>a or b

Keywords: number theory, integer, numerical, collatz