Negative Prime Numbers... TRUE or FALSE ???

Hello everybody,

It seems generally admitted that there are no negative prime numbers.

What are the rules that can affirm this?

Thanks in advance and happy new year to all.

Best regards,
Ex0 SySCreationAsked:
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Ryan ChongSoftware Team LeadCommented:
yes, there is no negative prime numbers. you could have read the definition at:

A natural number .. is ... if it has exactly two positive divisors, 1 and the number itself

Prime number
Ex0 SySCreationAuthor Commented:
Thanks for your so fast answer.

Wikipedia said : A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.......

In this case, why it would be wrong to said that "-585968374389626759" is a prime number ?

If we follow this common rule, this number is then divisible by 1 (positive) and of course itself (negative).

-585968374389626759  /  1  =  -585968374389626759
-585968374389626759  /  -585968374389626759  =  1

Is this correct ?
Ryan ChongSoftware Team LeadCommented:
if the definition of a prime number must be positive, then even -585968374389626759 can only be divisible by 1 and by itself, but the fact it's not a positive value, so it cannot be considered as a prime number.

for the equations:

-585968374389626759  /  1  =  -585968374389626759
-585968374389626759  /  -585968374389626759  =  1

Open in new window

these is correct to prove that the big number can be divisible by 1 and by itself.

some other discussions can be found here:

one interesting note is that if a prime number like: -7 can be considered as a prime number, but in the fact it's divisible by -1 and 7, and then 1 and -7.
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Ex0 SySCreationAuthor Commented:
Of course you can check that "585968374389626759" (positive) is a prime number on "" or then on ""

This is true that we can't check this same negative value on those websites.

Just have a look to our online tool that make it possible check negative prime numbers.

Verification of a negative value :
Ex0-Prime-Toolbar - Verification of a negative value
Next negative value :
Ex0-Prime-Toolbar - Next negative value
Previous negative value :
Ex0-Prime-Toolbar - Previous negative value

Do you think we made a mistake ?

Thanks in advance .. ... .....

P.S. We don not use any database, the process is always computed in live by our algorithm...
ste5anSenior DeveloperCommented:
In this case, why it would be wrong to said that "-585968374389626759" is a prime number ?
Simply by definition.
Cause  -585968374389626759 is NOT a natural number. Thus the prime number rules does not apply.

And now it's getting to be fun: You can generalize this idea. But then this is called prime element..
The full sentence of the Wikipedia definition is: "A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself."  The "greater than 1" excludes negative numbers.

It also fails the "no positive divisors other..." on two counts.  -585968374389626759 is not a positive divisor so that isn't part of "1 and itself".  It does have a second positive divisor, namely 585968374389626759, but that is neither 1 nor itself.
If you extend the definition of primes in this trivial way, so that -P is prime whenever +P is, then you break
     Fundamental Theorem of Arithmetic

In simple terms, if 2 is prime, then so is -2.  So 4 (the smallest composite number) no longer has a unique factorization.  In fact, no composite number will have a unique factorization.

This would do enormous damage to Number Theory with no discernible benefit.

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I think that d-glitch's comment alludes to an important point.  Prime numbers are prime number by definition, not by "discovery".  That is, they could have been defined in a number of ways (including those that allow negative primes), but their usefulness would have diminished.  Their properties are not intrinsic in the numbers themselves.  Rather, they are part of the model that is created by the definition.  That pretty much holds for nearly all (?) of mathematics.
Ex0 SySCreationAuthor Commented:
Thank you all for these very good answers.

So, finally, a prime number seems to be prime not because of its own intrasecs properties, but only by the definition of what is a "natural number" which is defined as only positive.

In this case, how to name all negative numbers mirrors to positive prime numbers that have the same values?

Is there a category for classifying these "negative prime numbers"?

Thank you for your reply and see you soon ... ... .....

Best regards,
"how to name all negative numbers mirrors"
I think you'll have to come up with a specific definition for this before it can be answered.

Depending on how you define "negative prime numbers" I would expect that there are none or there are exactly as many as prime numbers.

For example, 3 is prime.  -3 has 4 divisors: (+-1 and +-3).  If you eliminate the explicit restriction on primes being positive, negatives of prime numbers won't be prime as they'll have 4 divisors.  You could define them as "numbers that have only four divisors, positive and negative one and the positive and negative value of themselves", but what would be the point?

As d-glitch mentioned, this could well have far-reaching impact in other parts of Number Theory.

You make a reference to "intrinsic properties" of numbers.  I'm not sure that numbers have ANY intrinsic properties other than what arise from what we define.  0 and 1 may be exceptions to that, but I'm not even sure there.
ste5anSenior DeveloperCommented:
This is just a comment:

In this case, how to name all negative numbers mirrors to positive prime numbers that have the same values?
You can name them exactly like that. Mirrors.

Is there a category for classifying these "negative prime numbers"?
Afaik no. Cause the only generalization of prime numbers I know are prime elements, which require a (algebraic) ring. Z (all integers, positive, 0 and negative) form a ring, but primes require only one unit. Your approach over Z requires two. So strictly speaking, they are not "primes".

So CompProbSolv is right. Your definition maybe useful in your use-case, but in general terms this defintion has draw backs like violating the Fundamental Theorem.
I had to pick a best answer, but I intended an even, four-way split on the points.
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