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Dashin
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Complex Numbers have always facinated me.
But... Do complex primes exist? If so, How?
But... Do complex primes exist? If so, How?
Hurkyl said:There are no primes in the complex numbers. There are no primes in the real numbers either. When every non-zero number is invertible, there is no such thing as a prime!
In the integers, there are primes. There are primes in the gaussian integers as well. The gaussian integers are numbers of the form a + bi, where a and b are both integers.
In the gaussian integers, 2 is not prime; its prime factorization is (1-i)(1+i).
Dashin said:Complex Numbers have always facinated me.
But... Do complex primes exist? If so, How?
chiro said:Not really but you could define an analog in terms of behavior. The thing you would need to think about are what the atoms are in the complex case and the constraints.
Did you have an idea of something in mind?
Dashin said:Thanks!
So... What primes are there in the Gaussian Integers?
Hurkyl said:If p is an integer prime, and you can write [itex]a^2 + b^2 = |p|[/itex], then both [itex]a+bi[/itex] and [itex]a-bi[/itex] are Gaussian integer primes. If you cannot, then [itex]p[/itex] is also a Gaussian integer prime.
All primes in the Gaussian integers are of this form.
Doesn't matter; they just differ by a unit. Just like -5 and 5 are both integer primes -- and in a certain sense the "same" prime, (3+2i), i(3+2i), (-1)(3+2i), and (-i)(3+2i) are all the "same" gaussian integer prime.The prime factorization of 2 I mentioned earlier: I could have (and probably should have) also written it as (-i) (1+i)^2, since 1+i and 1-i are the "same" prime.Dashin said:Does the order of a and b matter? Do you need to make the larger number be a, or doesn't it matter?
Hurkyl said:Doesn't matter; they just differ by a unit. Just like -5 and 5 are both integer primes -- and in a certain sense the "same" prime, (3+2i), i(3+2i), (-1)(3+2i), and (-i)(3+2i) are all the "same" gaussian integer prime.
The prime factorization of 2 I mentioned earlier: I could have (and probably should have) also written it as (-i) (1+i)^2, since 1+i and 1-i are the "same" prime.
(Just like prime factorizations in the integers can have a (-1) out front, factorizations in the guassian integers can have a (-1), (-i), or i out front)
If you don't like multiple primes being the "same", then I suppose you could insist on a being positive, and being larger in magnitude than b. (and have a special rule for deciding which of 1+i, 1-i, -1+i, and -1-i you like)
chiro said:Not really but you could define an analog in terms of behavior. The thing you would need to think about are what the atoms are in the complex case and the constraints.
Did you have an idea of something in mind?
This is a commonly asked question in the field of mathematics and number theory. The answer is yes - complex primes do exist. A complex number is considered prime if it cannot be expressed as the product of two non-zero complex numbers.
Yes, complex numbers can be prime. Just like real numbers, complex numbers can also be prime if they cannot be divided evenly by any other number.
Complex primes, also known as Gaussian primes, have the same properties as real primes in that they cannot be divided evenly by any other number. However, they exist in the complex plane and have both a real and imaginary component, whereas real primes only exist on the real number line.
Complex primes have many applications in mathematics, particularly in number theory and cryptography. They play a crucial role in constructing complex numbers and have applications in solving problems related to factorization and encryption.
The Riemann hypothesis is a famous unsolved problem in mathematics that deals with the distribution of prime numbers. The existence and properties of complex primes are closely related to this hypothesis, as it involves analyzing the distribution of prime numbers in the complex plane.