Dear Shahid Nawaz

Natural numbers come first equipped with the basic operations of addition, subtraction, multiplication, and division. See Peano axioms on N.

Then we present the factorization of a given number due to the closure of the multiplication operation. Consequently, we define prime numbers that are greater than one with exactly two factors: 1 and the number itself.

To reverse the logical flow of such ideas will show many difficulties with the axiomatic model of numbers.

The fundamental theorem of arithmetic is not enough to create the whole axiomatic system of natural numbers. One needs more facts and axioms to do so.

Best regards

there is archeological evidence of natural numbers for many thousands of years; they come first

as a mathematical structure, each one can be decomposed into primes, which makes primes primary (logically), but only if you are assuming multiplication is the main operation; if it is addition, then only 0 and 1 are primary

actually, the term ``first'' is ambiguous; it could mean historically, logically, in importance, or many other things

The birth of natural numbers is the birth of Mathematics. You can refer the book, " Mathematics, its magic and mastery". A statement in it says, "A notch and Mathematics is born". It is the earliest instance that when a early man, even before learning to talk had used Mathematics to keep the track of his animals going out in the morning and coming back in the evening.

Prime numbers have come later.

Logically, natural numbers come first. The human used them to account or specify somethings around him. May be, he dealt with them without feeling. I think primer numbers came more late than them because primer numbers is a rank of thought higher than natural numbers. Investigations of primer numbers began when trends started towards mathematics.

of course, the integers are not a group under multiplication, only a semi-group, but peter's point is well-taken; we seem to be fascinated by foundations and inductive constructions based on them

but look up peter aczel's (non-well-founded set theory) book, and the category theoretical notion of co-induction: a marvelous paper by joseph goguen (circular co-inductive rewriting) shows that sometimes circular definitions produce sensible and well-defined mathematical objects, and co-induction is the key to making that work consistently

these are both fairly new math - aczel's book is 1977

what p.aczel does is to change the foundation axiom (any chain of memberships of the form: a has an element b, which has an element c, ... must be finite) to what he calls an anti-foundation axiom, and shows that the new set theory also contains the old one (ZF i think) as a subtheory - i.e., this new theory contains all of the sets we like and a bunch more besides (so it is non-well-founded, but not ill-founded 8-)) - he also provides a model of the new one contained in the old one, which is part of his proof of mutual consistency

it is a hard read, but there is a wonderful explanation in a short book by jon barwise and john etchemendy called ``the liar: essays on truth and circularity'', which is far more clear

co-induction is the dual concept to induction, as is fairly conventional in category theory: instead of starting at the bottom with a set of objects and building structures, you start at the top and decompose them into structures that may contain those same objects (the finite structures are the same, but there can be infinite structures in a co-inductive construction - the situation is essentially the same as the ``infinite'' lists in the programming language LISP, which are represented as an ordinary finite list of elements and a generator that makes more if you try to access them) - wikipedia has a pretty good introduction and references to several tutorials

so instead of constructing, you are unraveling, and you frequently do not have to unravel to the bottom to prove useful things - that is where the circular definitions can help (there are several instances on the web, plus a lot in the ucsd sites) - joe goguen was the only person i ever met who could make category theory look like a computational device (look for the programming languages maude and bobj)

Although we are talking in big terms, it is very simple.

What are prime numbers? If we go to the definition, it is clear that P >1 is the first restriction. What is 1? it is the first natural number and so out of natural numbers we are selecting the prime numbers which satisfy a certain property; i.e they are divisible by themselves or (+/-) 1.

So, it is very very clear that natural numbers are predecessors of prime numbers.