As always, at the beginning of the year a lot of people write about different properties of the number of the year (I did something similar last year in my 2016 Resolutions post). This year, I have already read a couple of great articles on the properties of 2017 and something was stuck in my head about 2017 being a prime sometimes.
As I tell my students, a number is prime if it is bigger than 1 and has just 2 factors (1 and itself). Taking this into consideration, 2017 is a prime number. Besides this, 2017 has many interesting forms that mathematicians have studied a lot. Here are some:
- 4n + 1 because 2017 = 4 × 504 + 1 and there are infinitely many primes of this form
- x² + (y²)² because 2017 = 44² + (3²)² and there are infinitely many primes of this form (proved in 1997 by Friedlander and Iwaniec). These are called the Friendlander – Iwaniec primes. Also, from this form you can see that 2017 can be written as a sum of two sqaures, which in fact is a conclusion from the above notation (4n + 1) by using Fermat’s 2 square theorem.
- x³ + 2y³ because 2017 = 11³ + 2×7³ and there are (again) infinitely many primes of this form (proved by Heath – Brown in 2001).
The thing that got my attention was not this list (which is longer than what I have put here) of forms of this prime number. What caught my attention was the fact that if we change the number system, this is not a prime anymore. Maybe that doesn’t sound extremely fantastic for you, but it got stuck in my head for a little while. This is because it reminded me of Group Theory and Galois Theory. Taking any set and an operation on that set, it was always interesting to find out the prime numbers of for that specific group, field or ring. Also, when I got to Galois Theory and solving polynomials, every time I had to factorise something was like a puzzle for me. This fact of changing the number system we are referring to is quite interesting.
From this point of view, any prime number which can be written as a sum of two square numbers will never be a prime number if we think about Gaussian integers, which are about complex numbers. A Gaussian integer is a complex number of the form a + b × i, where a and b are integers, and i is the imaginary unit ( i² = -1). This is because x² + y² = x
Therefore, just this small thing on finding different properties of a number can get us to an interesting road. As with any other real life problem, when we change the “system” in which we consider the problem we could find even more solutions, or our solutions could be completely wrong. This taught me that every problem we have, it is always good to see it from different angles and even from other people’s perspective. Something which could be a solution for us, could not be for someone else. Thinking about my example, a number could be prime in a number system and a composite number in other number system. So, for 2017, I wish you could stop for a second before blaming someone or something. Just change the “number system” and see how the problem (and the solution) changes. Enjoy every second of 2017.
Lots of love and don’t forget that maths is everywhere! Enjoy!