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shabbir 3Aug2009 20:33

Can negative numbers be prime ?
 
In Daily Article competition ( Sum of Prime Numbers | 3 Aug 2009 ) we had a debate which I would like to carry it here.

Can negative numbers be prime ?

According to me.

No
  1. By definition it has to be whole numbers greater than 1.
  2. A number is prime if its divisible by 1 and itself but negative numbers are always divisible by 1, -1 and itself.

Raj08 3Aug2009 21:46

Re: Can negative numbers be prime ?
 
I do agree with you shabbir. I don't think negative numbers can be prime.

rik625 3Aug2009 22:21

Re: Can negative numbers be prime ?
 
I say negative numbers can be prime too. Although we never consider this fact in basic calculations. But i do agree with SwasatPadhi.

mayjune 3Aug2009 23:34

Re: Can negative numbers be prime ?
 
1 Attachment(s)
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 :)

mayjune 4Aug2009 17:56

Re: Can negative numbers be prime ?
 
after all i have searched, i would say no it doesnt exist, simply for the reason i understood,
whole idea of prime is to factorize every number into its prime, if we have -ve primes, the concept of having prime itself will fall, as then there is no unique factorization....

read this "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."

Defintion of prime (wiki) :-
The fundamental theorem of arithmetic establishes the central role of primes in number theory: any nonzero natural number n can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.

Hence, we cannot have -ve prime numbers...

SaswatPadhi 4Aug2009 18:56

Re: Can negative numbers be prime ?
 
/ˉ\
#|
I don't agree to the above conclusion.

Because, the primes -2 and 2 are not two different primes ... -2 and 2 are exactly the same primes. Did you post this, without reading it ?? --

Quote:

Originally Posted by mayjune (Post 54157)
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.

(Relevant portions in red and bold)

So, ... say the number 4 has ONLY ONE unique representation of prime factorization. You may write is as 2^2 or (-2)^2 doesn't matter. They are the same.

It's like having a reference variable in C++.
If a is a reference to b, then you can use a instead of b and that won't matter 'cuz a is an alias (note this word) of b.
So, -p is an alias of the prime p.

I would like to mention something here, that I very much liked :

Quote:

You have to get a feel for what mathematics is about. It's not about cranking numbers through equations and getting answers, and it's not about balancing your checkbook or figuring out how long it will take Bill and Janet to mow the lawn if they work together, and it's not about building bridges or telephones or sending spaceships to other planets. Those are all _uses_ of math, but mathematics itself is about searching for patterns.
Source : http://mathforum.org/library/drmath/view/55940.html (Original)
######http://74.125.153.132/search?q=cache...ient=firefox-a (Google Cache, 'coz I could not access the original)

SaswatPadhi 4Aug2009 18:58

Re: Can negative numbers be prime ?
 
Assuming, shabbir is yet to announce the results; we can continue this further.

mayjune 4Aug2009 19:46

Re: Can negative numbers be prime ?
 
it can go on and on...i think we should keep that puzzle as no winners....

SaswatPadhi 4Aug2009 20:11

Re: Can negative numbers be prime ?
 
nah .. let Kshiteej win, 'coz votes in favor of my view is less.

mayjune 4Aug2009 20:12

Re: Can negative numbers be prime ?
 
well it still is a debatable topic ...so i don't think 6 votes can decide that...
i think it should be no win, let him decide...


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