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 By definition it has to be whole numbers greater than 1. A number is prime if its divisible by 1 and itself but negative numbers are always divisible by 1, -1 and itself.
I say negative numbers can be prime too. Although we never consider this fact in basic calculations. But i do agree with SwasatPadhi.
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
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...
/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 ?? -- (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 : Source : http://mathforum.org/library/drmath/view/55940.html (Original) ######http://74.125.153.132/search?q=cach...mes&cd=1&hl=en&ct=clnk&gl=in&client=firefox-a (Google Cache, 'coz I could not access the original)
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...
No, that means Kshiteej wins ! Check this : http://www.go4expert.com/showpost.php?p=54234&postcount=10
This is just about definitions so the only possible answer can be "no". For a "Yes" answer the definitions must be changed. http://en.wikipedia.org/wiki/Prime_number states "a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself" and http://en.wikipedia.org/wiki/Natural_number states "there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition or the set of non-negative integers {0, 1, 2, ...}" - so this excludes all negative numbers from the definition of a prime. and http://en.wikipedia.org/wiki/Divisor states "a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder." Also if primes are redefined to include x<0, why is that of any interest? If x is prime then -x is also prime, so adding -2, -3, -5, -7 etc to the list adds no new information. Or you end up with no prime numbers at all, if for example -5 is divisible by -5, -1, 1 and 5, so every prime number by the current definition is not prime because any number x is divisible by +x, -x, +1 and -1. Or you extend the definition to say "exactly four distinct divisors: +/- 1 and +/- itself", in which case you're back to a set of primes where +x and -x are both prime or both not prime and so it has no benefit over the positive numbers only definition as is currently the case.
Wow, this thread is still alive and kicking !! :rofl: But, I request all members to give a justification for their vote, without which it's meaningless. If you agree with someone's views like xpi0t0s, mayjune or shabbir (all of them have almost similar reasons, though :p); just mention their name .. no need to write the same justification again