It is quite well-known that any positive integer can be factored into a product of primes in a unique way, up to an order. (And, that 1 is neither prime nor composite) —- we all know this from our high school practice of “tree-method” of prime factorization, and related stuff like Sieve of Eratosthenes. But, it is so obvious, and so why it call it a theorem, that too “fundamental” and yet it seems it does not require a proof. It was none other than the prince of mathematicians of yore, Carl Friedrich Gauss, who had written a proof to it. It DOES require a proof — there are some counter example(s). Below is one, which I culled for my students:
(a) Show that the sum and product of elements of E are in E.
(b) Define the norm of an element by . We say that an element is prime if it is impossible to write with , and , ; we say that it is composite if it is not prime. Show that in E, 3 is a prime number and 29 is a composite number.
(c) Show that the factorization of 9 in E is not unique.