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So I recently had a maths lesson about prime numbers at school and got bored. So I started to mess around with prime numbers beyond what we were doing, in particular negative numbers and primes.

I came to a few conclusions:

Note: I am defining a prime as

An integer with only 2 real divisors: 1 and itself

1) No negative numbers are prime as they all have more then 2 divisors: (for $x$ when $x$ is a negative integer)

  • 1
  • -1
  • $x$
  • -$x$

2) There is only 1 prime that satisfy the above definition:

$-1$

Every other integer (except 1) has at least 4 divisors; the ones stated above.

$1$ has 2 divisors but doesn't satisfy the above definition as its divisors are 1 and -1 which are not "1 and itself"

Therefore, the only number that satisfies this definition is -1!

In case you don't believe me, here are some examples

With $x$ as 11:

Divisors:

  • $1 : 11 / 1 = 11$
  • $11 : 11 / 11 = 1$
  • $-1 : 11 / -1 = -11$
  • $-11 : 11 / -11 = -1$

With $x$ as 25

  • $1 : 25 / 1 = 25$
  • $5 : 25 / 5 = 5$
  • $25 : 25 / 25 = 1$
  • $-1 : 25 / -1 = -25$
  • $-5 : 25 / -5 = -5$
  • $-25 : 25 / -25 = -1$

Clearly I've done something wrong but I'm not sure exactly what it is. Can anyone help? Have I accidentally discovered the greatest conspiracy theory in mathematics?

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closed as off-topic by YiFan, Lord Shark the Unknown, Leucippus, José Carlos Santos, GNUSupporter 8964民主女神 地下教會 Mar 1 at 9:18

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    $\begingroup$ Uhh. Primes are only non- negative integers. $\endgroup$ – The Dead Legend Apr 21 '17 at 22:54
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    $\begingroup$ You can define a prime number that way, but don't be surprised if others don't follow you in droves. The reason why prime numbers are defined as those natural numbers other than $1$ that are divisible only by $1$ and themselves is because they're interesting that way, not because that's the "purer" way to define them or anything like that. Although your reasoning is basically sound, I don't see much that's interesting about a set of numbers whose only member is $-1$. $\endgroup$ – Brian Tung Apr 21 '17 at 22:55
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    $\begingroup$ I can do you one better. If I define a prime number as a pink elephant wearing a tutu that can do differential calculus while traveling backwards in time, then there are NO prime numbers. Take that! ... So "An integer with only 2 real divisors: 1 and itself" well, obviously there are no such numbers. $\endgroup$ – fleablood Apr 21 '17 at 22:55
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    $\begingroup$ People: this is a non-issue. The question is not "what is the definition" (that appears to be open to discussion, indeed), but "what is a useful definition" (or, better, "characterization"). E.g., to "define" (if it were within one's power) "primes" to be such-and-such and thing of which there are nearly none... would be both uninteresting and useless. In particular, there is "no contest" about "definitions". The popular mythology that there is a quasi-sacred "definition" of everything is wildly incorrect. Good mathematicians mostly attempt to talk about palpably realy/useful things. $\endgroup$ – paul garrett Apr 21 '17 at 23:10
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    $\begingroup$ Nice proof of Riemann hypothesis ! I am very curious of further applications of your theorem to cryptography $\endgroup$ – user171326 Apr 22 '17 at 0:30
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Have I accidentally discovered the greatest conspiracy theory in mathematics?

No.

Your definition of "prime number" is not how prime numbers are actually defined. While it is common to say "A number is prime iff it has no divisors other than itself and $1$," this is somewhat imprecise: the actual definition of a prime number is "A positive integer $>1$ with no positive integer divisors other than $1$ and itself."

It is sometimes useful to consider variations of this definition - the most common one being to allow negative primes as well, and here a prime is any integer which generates a nonzero prime ideal - but this is the standard definition. Showing that your own definition behaves differently, just shows that your own definition isn't equivalent to the standard one.

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  • $\begingroup$ The quote is just a jokey footnote. Don't take it too seriously. $\endgroup$ – caird coinheringaahing Apr 21 '17 at 23:03
  • $\begingroup$ @ThisGuy I know it's a joke, I just thought it made a good way to begin my answer. $\endgroup$ – Noah Schweber Apr 21 '17 at 23:04
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    $\begingroup$ +1 fpr the last sentence. I'd even put that one first (or right after the joke). Mathematicians define things precisely only after they understand intuitively what it is they are trying to define. $\endgroup$ – Ethan Bolker Apr 22 '17 at 0:00
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It's true that, if you admit negative integers, every integer other than $1$ and $-1$ has at least $4$ divisors. However, there are good reasons that we do not define these to be prime. In general, the important bit about a definition of a prime is that we ignore what are called units.

To start with, the multiplicative identity of the integers is $1$ because it $1\cdot x = x=x\cdot 1$. That is, multiplying by it does nothing. We say that an integer $x$ is a unit if there is some other integer $y$ with $xy=1$. You can convince yourself that the only units in the integers are $1$ and $-1$. Equivalently, a integer is a unit if it divides every other integer.

For this reason, one defines an irreducible integer as an integer $x$ with the property that if $x=yz$ for integers $y$ and $z$, then one of $y$ or $z$ is a unit. That is, we are ignoring divisors that are units, because they're uninteresting - they divide everything! For the integers, these are exactly the elements called prime for various reasons - but you can convince yourself that the irreducible elements are the primes and their negatives.

In higher mathematics, one often replaces "integers" in the above discussion with another more general object called a ring. If you are familiar with complex numbers, an interesting exercise is to repeat all of this discussion where you think of numbers of the form $a+bi$ for integer $a$ and $b$ rather than just the integers. You can also redo the discussion using rational or real numbers, in which can you realize that everything is a unit (except $0$).

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That's a pretty cool discovery, but I doubt we are going to change our definition of prime on the basis of it. Of course, sometimes we do change our definitions: for a while, 1 was considered a prime, but at some point we decided it was no longer a prime. For one thing, with 1 no longer being a prime we obtained a very nice fundamental theorem of arithmetic, and other goodies. But in your case, we are not getting a lot of goodies. Indeed, how useful would a concept be if there is only one object that fits that concept?! But again, I congratulate you on your discovery, and I would encourage you to play around more with things like this ... It's how discoveries are made.

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