The Fundamental Theorem of Arithmetic states that every composite integer can be written as a unique product of prime numbers. As new prime numbers are being discovered, mathematicians continue their search for larger and larger prime numbers. How do mathematicians find large prime numbers? They'll use what's known as a primality test which is an algorithm that will determine whether a number is composite or prime. By using these primality test algorithms, mathematicians can use computers to find massive prime numbers that would normally take several years to determine. However, not all primality tests are perfect. Deterministic primality tests are algorithms from which we can always conclude a number is composite or prime. Probabilistic primality test are algorithms that gives us hope that a number could be prime (or composite) but there is not a definite conclusion.
Chu, H. (2018). Prime Numbers and Primality Testing (Undergraduate honors thesis, University of Redlands). Retrieved from https://inspire.redlands.edu/cas_honors/178