Factoring 5

We grow up in school learning about the expanding world of numbers. It's fun to count to 300 in the bath, and as we progress to fractions, negative numbers, and irrational numbers, we feel a sense of enjoyment as the things we can do steadily increase. Many of you were probably satisfied to learn that with complex numbers, any quadratic equation can be factored into the product of two linear expressions:

And, of course, while a+ib and a−ib are no longer ordinary integers, we can take a leap of imagination and think of them as evolved integers, formed by combining the ordinary integers a and b with i. This naturally leads to the ever-expanding question: "What would happen if we reconsidered prime factorization using a clever new type of integer?" This is clearly a more insightful approach than just blindly searching for an answer. I learned this as a student, and while I was disappointed that I hadn't come up with it myself, it was still one of those joyful moments when I felt the freedom of mathematics (after all, we are prime factoring a prime number).

This might be a bit advanced, but the answer in this case is also quite remarkable. In fact, the set of all primes we are looking for

is exactly the same as the set of all prime numbers that appear in the prime factorization of, which is

precisely the set of all prime numbers that leave a remainder of 1 when divided by 4,