Fundamental Theorem of Arithmetic

Prove that every integer greater than 1 can be written as the product of primes.

The problem at hand is a foundational result in number theory often referred to as the Fundamental Theorem of Arithmetic. It states that every integer greater than one is either a prime number itself or can be expressed as a product of prime numbers, and moreover, this representation is unique, up to the order of the prime factors. The significance of this theorem is immense, as it forms the bedrock upon which the structure of integers is understood.

When approaching this problem, a key concept to utilize is mathematical induction, a powerful proof technique used to establish the truth of an infinite sequence of propositions. Lay the groundwork by showing the base case is true, which is generally the smallest integer greater than one, such as two or three. Next, assume the proposition holds for some integer k and then demonstrate it is true for k+1. This leap reinforces the structural foundation of integers regarding prime composition.

Additionally, the problem indirectly emphasizes understanding what prime numbers are: those numerically indivisible by numbers other than one and itself. Recognizing primes and decomposing integers into their prime factors are skills that not only prove this theorem but also underpin various algorithms and applications in cryptography and computer science.