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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?

caird coinheringaahing
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  • I suggest googling 'prime number'. Reading should put the controversy to rest. – user121330 Apr 21 '17 at 22:54
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    Uhh. Primes are only non- negative integers. – The Dead Legend Apr 21 '17 at 22:54
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    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 $-1$. – Brian Tung Apr 21 '17 at 22:55
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    If you want to talk about real divisors, you might as well say that for example $11$ is divisible by $\pi$, since $11=\frac{11}{\pi}$. Therefore there are no primes! – Arnaud D. Apr 21 '17 at 22:55
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    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. – fleablood Apr 21 '17 at 22:55
  • Is it really so easy to prove that there are no prime integers that are also elephants wearing tutus? – Andrew Dudzik Apr 21 '17 at 22:59
  • @Slade: I believe that fleablood's definition forces prime integers to be elephants wearing tutus. But not all tutu-wearing elephants are prime integers; they must also be able to differential calculus while travelling backwards in time. That they don't exist turns out to be quite difficult to prove, since just because the elephants don't do it doesn't mean they can't. They might just not want to. It's non-trivial to show that they have no more status than the Frey curve. – Brian Tung Apr 21 '17 at 23:03
  • Well, 3 wears tutus all the time but she can't do differential calculus. Seven can but he is stationary in time. 11 can is green, not pink. 17 could do all of those but he is really hung up on gender issues and refuses to wear a tutu even if it would prove me wrong. – fleablood Apr 21 '17 at 23:03
  • @fleablood actually my pet elephant in 2050 might disagree... – caird coinheringaahing Apr 21 '17 at 23:04
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    Are you claiming that primes can't be defined in such a way as to give the usual set of numbers considered prime? Or that they're often defined incorrectly/imprecisely? Or are you asking what's wrong with your definition? – pjs36 Apr 21 '17 at 23:05
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    But seriously, for the OP: Although there are interesting definitional questions in mathematics (for instance, is the continuum hypothesis "true"? you get some interesting answers to this), you'll have to show that some interesting things happen by defining prime numbers this way. Otherwise, it's just some weird corner case in the way language works when dealing with mathematical precision. – Brian Tung Apr 21 '17 at 23:05
  • 1 has 5 and 1/5 as factors so 1 is not prime either. – fleablood Apr 21 '17 at 23:06
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    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. – paul garrett Apr 21 '17 at 23:10
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    Nice proof of Riemann hypothesis ! I am very curious of further applications of your theorem to cryptography –  Apr 22 '17 at 00:30
  • Maybe not a duplicate, but related : https://math.stackexchange.com/questions/1645111/1-as-the-only-negative-prime?rq=1 – Arnaud D. Apr 22 '17 at 18:11

3 Answers3

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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.

Noah Schweber
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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 $0$).

Milo Brandt
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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.

Bram28
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