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**General Science / Re: Is 2 really prime? If so, why isn't 1?**

« **on:**30/04/2022 00:19:02 »

Hi.

There is already some terminology you could use to describe a set of things that behave like prime numbers but apply to a much more generalised set of objects than just the Natural numbers. These things are called "prime elements" and the parent algebraic structure is known as a "Ring". You seem to be interested in the Ring which is the Integers (positive and negative Naturals with 0, under conventional binary operations of + and x).

See Wikipedia entry: Prime elements, if you're interested. https://en.wikipedia.org/wiki/Prime_element

However, you should note that they exclude "units" which would include -1 in the ring of Integers, i.e. they would directly exclude both +1 or -1 from the prime elements in the ring of Integers. (For what reason? Similar to excluding 1 from the primes, it makes it much easier to state an equivalent unique factorisation theorem for the ring of Integers).

Here's a quick question or puzzle, just for fun. It relates to the idea you mentioned earlier of eliminating the number 2 from the prime numbers. You also seemed keen to extend beyond the positive numbers and consider negative numbers but you really don't have to stop there - you can consider Complex integers.

The Complex Integers or "Gaussian Integers" are the Complex number equivalents of integers. Specifically, the Gaussian integers are the set of all complex numbers of the form a+bi where a and b are integers.

Just like in ordinary arithmetic with Natural numbers, a prime (or prime element) of the Gaussian integers is a Gaussian integer, p, that is irreducible or cannot be factorised. Specifically, if we have p = q × r (where × is just ordinary multiplication of the complex integers q and r) then at least one of q or r must be a unit element. A "unit" is any complex number that lies on a unit circle around 0, so the only units in the Gaussian integers are +1, -1, +i, -i.

The number 2 is a prime in the ordinary integers. Is it still a prime in the Gaussian integers? To say that another way, can you factorise the number 2 in the Gaussian integers?

Best Wishes.

I came across another wrinkle: what about –1?Your general arguments after this are reasonable. However, I think it is again just a matter of simplicity and having a set of numbers that are useful for something. It is possible and useful to confine your attention to what people might call the counting numbers or the Natural Numbers, so we do. That doesn't mean that mathematicians have never considered generalising the idea of prime numbers and investigating properties like prime factorisation in a structure bigger or more abstract than just the positive counting numbers - they certainly have.

There is already some terminology you could use to describe a set of things that behave like prime numbers but apply to a much more generalised set of objects than just the Natural numbers. These things are called "prime elements" and the parent algebraic structure is known as a "Ring". You seem to be interested in the Ring which is the Integers (positive and negative Naturals with 0, under conventional binary operations of + and x).

See Wikipedia entry: Prime elements, if you're interested. https://en.wikipedia.org/wiki/Prime_element

However, you should note that they exclude "units" which would include -1 in the ring of Integers, i.e. they would directly exclude both +1 or -1 from the prime elements in the ring of Integers. (For what reason? Similar to excluding 1 from the primes, it makes it much easier to state an equivalent unique factorisation theorem for the ring of Integers).

Here's a quick question or puzzle, just for fun. It relates to the idea you mentioned earlier of eliminating the number 2 from the prime numbers. You also seemed keen to extend beyond the positive numbers and consider negative numbers but you really don't have to stop there - you can consider Complex integers.

The Complex Integers or "Gaussian Integers" are the Complex number equivalents of integers. Specifically, the Gaussian integers are the set of all complex numbers of the form a+bi where a and b are integers.

Just like in ordinary arithmetic with Natural numbers, a prime (or prime element) of the Gaussian integers is a Gaussian integer, p, that is irreducible or cannot be factorised. Specifically, if we have p = q × r (where × is just ordinary multiplication of the complex integers q and r) then at least one of q or r must be a unit element. A "unit" is any complex number that lies on a unit circle around 0, so the only units in the Gaussian integers are +1, -1, +i, -i.

The number 2 is a prime in the ordinary integers. Is it still a prime in the Gaussian integers? To say that another way, can you factorise the number 2 in the Gaussian integers?

**Spoiler:**show

Best Wishes.

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