Abstract

The sums of prime components of composite numbers lead to prime numbers, a remarkable phenomena in number theory that is examined in this paper. We present a constructive proof methodology to show that adding the prime components of the resulting composite numbers iteratively converges to prime numbers for any composite number higher than four. By applying logical reasoning and the fundamentals of number theory, we prove that all prime numbers larger than three have a finite number of "prime friends," or composite numbers whose prime factors add up to the prime number itself. We reveal the method by which prime numbers emerge from this iterative process by analyzing partitions of prime numbers and applying the function 2k + 3. This abstract summarizes our investigation into the convergence of prime numbers through the sums of prime factors of composite numbers, demonstrating a significant relationship between primes and composites in number theory. NOTE : We have created the term "prime friend"; it is not a term found in the traditional lexicon of mathematics.