The Real Algebraic Expression of the Collatz Conjecture

The Collatz conjecture is a very intriguing topic since a simple 3n+1 algebraic expression can create an endless 4 to 2 to 1 loop for any positive odd integer. Not surprisingly, the famous 3n+1 is not the only algebraic expression that can create an infinite loop when the same conditions for the Collatz conjecture is applied. This paper details the real algebraic expression discovered for the Collatz conjecture as well as the pattern of the infinite loops created by each specific algebraic expression. Section 2 reveals that the real algebraic expression of the Collatz conjecture is 3n+(3^k) and the loop follows the pattern of [1 x 3^k, 4 x 3^k, 2 x 3^k] for any positive odd integer, where k is any positive integer ranging from 0 to 31. Moreover, the algebraic expression of the Collatz conjecture for any negative odd integer is 3n-(3^k) and the loop follows the pattern of [1 x (-(3^k)), 4 x (-(3^k)), 2 x (-(3^k))], where k is any positive integer ranging from 0 to 31. Section 3 details the key to the pattern of the Collatz conjecture’s infinite loop, while section 4 and 5 provides proof for the key mentioned in section 3. Section 6 reviews the pattern of the infinite loops created in section 4 and 5. Section 7 provides an alternative method for determining the pattern of the infinite loop created by a specific algebraic expression of the Collatz conjecture using the key mentioned in section 3