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There are many possible definitions of a prime number such as the google search result of ‘prime number definition’:

’a number that is divisible only by itself and 1 (e.g. 2, 3, 5, 7, 11).’

Though by this definition, one could argue that therefore the only prime number is -1. Take 2 as an example, it is divisible by 2, 1, -1, and -2; as none of those give a fractional part. And if one argues that factors cannot be negative, that I see to say the negative numbers can’t have proper pairs of factors. -1 however I saw only divisible by 1 and -1 so it is the only integer that fits the definition.

There are several other definitions, that give our definition of prime numbers, but some seem over the top. So my question is as follows: Using mathematical notation, what is the fewest symbols the set of prime numbers (in the group of natural numbers bounded under addition) can be defined.

edit - Please read the actual question before commenting it as a duplicate or giving answers that don’t actually answer my question. This is not asking for a definition of prime numbers, I’m asking for the SIMPLEST POSSIBLE DEFINITION, as in, requires the least amount of information, of the set of prime numbers.

Logan M
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  • Comments are not for extended discussion; this conversation has been moved to chat. – Aloizio Macedo Oct 18 '18 at 16:44
  • It all depends on what you're using the set of primes as. If you need $\pm 1$ you simply union that in and define a new set. If you need all negative primes you can say $x$ is prime iff $(x)$ the ideal generated by $x$ is prime (in $\Bbb{Z})$. – Daniel Donnelly Oct 22 '18 at 21:23

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