This can go on and on....
but one basic problem i think there would be if we have prime numbers as -ve is, what is the smallest prime number? surely it can't be 2 if we consider -ve ones?
and i checked wolframalpha, it doesn't know -ve prime numbers...
This is an open debatable topic, some agree some disagree,
a large set of people say the definition of prime numbers was set before the -ve numbers, thus -ve numbers were not counted,
some argue
"t is not because primes were defined before negatives that negatives are not considered prime. Mathematicians easily adapted other concepts (evens/odds, for example) to negative numbers. Negatives are not considered prime because if they were, the Fundamental Theorem of Arithmetic (unique factorization into primes) would no longer be true. If -2 were prime, for example, we could 'prime factorize' 4 in two different ways: as 2^2 and as (-2)^2. If only positive numbers are considered prime, then every positive integer larger than 1 has one and only one prime factorization.

It is because the Fundamental Theorem of Arithmetic is truly fundamental to all of Number Theory that only positive numbers are considered prime."

some have there own concepts that it can be, ofcourse with various theories, some say it doesnt matter :-
"Answer One: No.

By the usual definition of prime for integers, negative integers can not be prime.

By this definition, primes are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded. In fact, they are given no thought.
Answer Two: Yes.

Now suppose we want to bring in the negative numbers: then -a divides b when every a does, so we treat them as essentially the same divisor. This happens because -1 divides 1, which in turn divides everything.

Numbers that divide one are called units. Two numbers a and b for which a is a unit times b are called associates. So the divisors a and -a of b above are associates.

In the same way, -3 and 3 are associates, and in a sense represent the same prime.

So yes, negative integers can be prime (when viewed this way). In fact the integer -p is prime whenever p, but since they are associates, we really do not have any new primes. Let's illustrate this with another example.

The Gaussian integers are the complex numbers a+bi where a and b are both integers. (Here i is the square root of -1). There are four units (integers that divide one) in this number system: 1, -1, i, and -i. So each prime has four associates.

It is possible to create a system in which each primes has infinitely many associates.
Answer Three: It doesn't matter

In more general number fields the confusion above disappears. That is because most of these fields are not principal ideal domains and primes then are represented by ideals, not individual elements. Looked at this way (-3), the set of all multiples of -3, is the same ideal as (3), the set of multiples of 3.

-3 and 3 then generate exactly the same prime ideal. "

i think the topic can go and on, its like discussing about infinity...
here's a nice joke on math i found...have fun
Attached Images
File Type: gif frazz.infinity.gif (27.1 KB, 5 views)
SaswatPadhi, shabbir likes this