Allow me to say that I am not sure if this has been explored previously. Let's say we have a number

, and we have to determine if the number is prime. We could either try and divide it with all the prime numbers known to us, or, as I realised some time ago, we could write it as the reult of an algebraic expression.

For example, let

be the difference of two squares,

and

. i.e.

It should be instantly clear that if

is not equal to

, then any positive number that can be expressed as such a difference is composite. If the difference between the two is one, then it appears to me that the only way to determine its nature is through brute prime factorisation.

Note:

.

I generalised this statement for algebraic expressions like

,

. It is even easier to prove that any number expressible as

is always composite, no matter the difference between

and

.

My reason for explaining all this is because, after trying unsuccessfully to try and find a suitable identity for

, it strikes me that there are some expressions which cannot be resolved into algebraic identities, and hence there will always be some numbers for whom the only method of determining their nature is brute prime factorisation.

My question, then, is this:

*why* are there some expressions which cannot be broken up into multiplications of two or more factors? Is there a general theory that allows us to determine if a given expression can have Diophantine expansion (my term - if there is another, more proper name, I would be delighted to know of it) i.e. where

and

are both whole numbers? If there is not, is there any related work on this subject which I may peruse?

Thanks.