DISCERNING NEW PATTERNS IN THE GENERALIZED COLLATZ FUNCTION UTILIZING BIG DATA ANALYSIS

The Collatz Conjecture has eluded mathematicians for decades. The small fact that for any initial number n, when recursively inputted to f(n)=n/2 if n is even and f(n)=3n+1 if n is odd will converge to 1 is a point of debate in the field of math for almost 80 years. However, the Generalized Collatz Function is a relatively new function that is not studied by many. Using the logic f(n)=n/d if n=0(mod d) and f(n)=mn+1 if n is not equal to 0(mod d), where d is a list of consecutive positive primes until the d^th prime, the Generalized Collatz is a variable form of the Classic Collatz. We use Mathematica to run the generalized Collatz sequence 1 million times and use big data scanning tools to create visualizations with high-probability to contain a pattern. We discern three new patterns from this big data. One, that all numbers that are divisible by a prime number squared can be set to n, so that, with m=n-1 , creates a hailstone sequence ending in {4, 2, 1}. Two, that all generalized Collatz sequences with m=n-1 will converge to 1, for any divisor, in three iterations. Three, that all Mersenne numbers, when set to m, will, for any divisor and for any initial number, cycle. This project, with the help of 3D graphing technology, was able to find never before seen patterns in a thought-to-be unpredictable function, the Generalized Collatz Function.