XIX - The Unique Prime Factorization Theorem

Abstract

In this chapter, we discuss our system's proof of the unique prime factorization theorem, also known as the fundamental theorem of arithmetic. This theorem is certainly the deepest and hardest theorem yet proved by our theorem-prover. The principal difficulty behind the proof is that Euclid's greatest common divisor function ( GCD ) plays an important role, even though it is not involved in the statement of the theorem. A beautiful but surprising fact (that multiplication distributes over GCD) is used; the more obvious fact that the GCD of two numbers divides both of them is also used. No other theorem yet proved by the theorem-prover employs as a lemma a surprising fact about a function not involved in the statement of the theorem.