Negative Prime Numbers? Yes, No, Maybe So

 The other day in class, one of my students asked me if prime numbers could be prime.  That question stopped me cold because I've never given that particular topic any thought.

The student thought if the definition of a prime is one and the number itself, then - 3 could work because it would be 1 and -3.  I told him, I'd investigate but I didn't think it would work that way.

After a bit of research, the answers I received were yes, no, and maybe depending on how you looked at things.  Now for the break down of all the answers.

1. I'll handle no because that is the one we are all familiar with.  According to this definition, all prime numbers must be greater than one and negative numbers are less than one. Also the definition is based on natural numbers and natural numbers do not include negative numbers so again they have to be positive.  This definition was made back in the time of the Greeks who only dealt with positive numbers.

Based on an activity I read about, negative prime numbers would not work.  Using manipulative, prime numbers will always have one left over after arranging the rest in a rectangular or square shape.  All composite numbers can be arranged in a rectangular or square shape with no left overs.

2.  Yes, there are negative prime numbers but you have to go into a very special branch of mathematics for this to be true. If you assume that for the statement -a divides b when every a does, they are then treated as the same divisor.  The numbers that divide one are called units and the two numbers a and b for which a is a unit time b are referred to as associates.  So the divisors a and -a of b are associates.  This means that -5 and 5 are associates because they represent the same prime.
Basically, if you begin looking at numbers within a ring structure, you can have negative prime numbers.

3. Maybe so because as you've seen it is possible to have them in ring theory and a few other areas of math but does it really matter since most standardized tests use the normal accepted definition.  They don't usually focus on things like ring theories.

If you are a bit rusty on things, Ring theory is a set with only two operations, usually addition and multiplication that meet certain criteria such as additive and multiplicative identities, additive inverses, addition being commutative, and the operations are associative and distributive.  It came out of Algebraic Number Theory and are generalizations and extensions of integers and algebraic geometry.

Just a bit of background on ring theory since within that part of mathematics, there are negative prime numbers.  Let me know what you think, I'd love to hear.  Have a great day.