Write a research paper on how this set generates a field when you perform modular addition and multiplication.

Write a research paper on how this set generates a field when you perform modular addition and multiplication.This is a research project, and I am asking for a completed research paper; this was my original idea

“let p be your favorite prime number. Define the set Z_p = {0,1,2,3,4,…,p-1}. If you do modular addition and multiplication, this set forms something called a field. Now, define the following function: f(x) -3x+1 if x is odd. f(x) = “2 inverse” times x if x is even. I write 2 inverse instead of 1/2 because in Z_p, the solution a to 2a is not 1/2 in the traditional sense. This is the collatz map, but written in Z_p. We ignore 0 in Z_p. The question is if every number in Z_p eventually goes to 1 if we repeat f(x) over and over again . As an example, let p=5, so that Z_5={0,1,2,3,4}. The first step is to find out what 2 inverse is in Z_5. It turns out that 2 inverse is 3, since 2 times 3 = 6 which is 1 in Z_5. We ignore 0, and 1 of course is already at 1. 2 is even, so we do 2 inverse times 2 = 3 times 2 =1, so 2 works. 3 is odd, so we do 3 times 3+1 which is 0 in Z_5. And 0 gets stuck at itself, so 3 never goes to 1 in this process, so our collatz conjecture fails in Z_5. The question I propose is for what primes p, does the collatz conjecture as stated above work in Z_p?”

Attached is a google document of essentially an outline that I drew up, and some white board work. If you are also versed in java (the programming language), can you include a code that can essentially helps speed up the process of finding these primes and attached it to the research paper with some diagrams, but not too much that its a majority of the paper.