Abstract: We learned in school that prime numbers are numbers that only factor as 1 times themselves. However, this definition assumes that the only allowable factors are positive integers. What does it mean to be prime if we are allowed to write 5 = (-1) x (-5) or 5 = sqrt{5}^2 (where “sqrt” denotes the square root)? How can a prime number factor if we extend our allowable factors beyond the positive integers?

We also learned that every integer can be written as a product of powers of prime numbers in such a way that the prime numbers and their exponents are unique (only their order isn’t unique). For example, 6 = 2 x 3. But if we are also allowed to write 6 = (1+sqrt{-5})(1-sqrt{-5}), the factorization is no longer unique. So how can we re-achieve uniqueness if we again allow factors beyond positive integers?

This talk will explore answers to the above questions and more. No mathematical background beyond high school is required.