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The prime numbers are usually defined to be positive integers with exactly two distinct divisors—one and the number itself. There are plenty of variations on this definition.

Just out of sheer curiosity, is there some alternative definition of prime numbers? I mean something like: The $p$ is a prime number iff it can be expressed as (some expression), or iff some identity involving $p$ holds.

Does the knowledge that some number is a prime number imply that the other number $p$ must be a prime? In other words, the $p$ is a prime iff (some expression involving $p$) is a prime. For example, if $2^n - 1$ is prime then also the $n$ must be a prime. Can a similar property be used to define all prime numbers?

Unproven conjectures are also welcome.

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    $\begingroup$ There are many ways of thinking about what prime numbers are. On the first page of his article "The First 50 Million Prime Numbers," Zagier points out that primes are the characteristics of finite fields, the nonzero prime ideals in the integers, and they correspond to the non-archimedean valuations on the rational numbers ($p$-adic absolute value for each prime $p$). $\endgroup$
    – KCd
    Feb 15, 2019 at 13:19
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    $\begingroup$ With all due respect, alternate definition is a waste of time. If you are talking about the same set of prime numbers then all alternate definition will eventual lead you to the same set of number. So you can save you time, energy, money, paper and ink by just gong ahead with the simplest definition which anyways the normal accepted definition of primes. $\endgroup$
    – Nilotpal Sinha
    Feb 17, 2019 at 6:23
  • $\begingroup$ @NilotpalKantiSinha you can reformulate almost anything in math, it's what proofs are made of. $\endgroup$
    – user645636
    Mar 4, 2019 at 3:17

3 Answers 3

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There are many theorems of the form "such and such is true if and only if $p$ is prime." Every one of these theorms make a perfectly okay definition of the prime numbers, but they wouldn't motivate the definition at all; why would we want to study these objects that seem to be so arbitrary? Examples are Wilson's Theorem ($(p-1)!=-1$ in $\mathbb Z_p$ iff $p$ is prime), $\phi(p)=p-1$ if and only if $p$ is prime ($\phi$ is Euler's totient function), et cetera. I think the only example which does actually make a lot of sense as alternative definition of primes is this: $p$ is prime iff $p\mid ab$ implies $p\mid a$ or $p\mid b$. Indeed, in general rings, this is taken to be the definition of a prime element.

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    $\begingroup$ $x^p\equiv x \pmod p$ is not a sufficient condition; see Fermat pseudoprimes. $\endgroup$
    – hmakholm left over Monica
    Feb 15, 2019 at 11:51
  • $\begingroup$ @HenningMakholm I thought Femat pseudoprimes to base $a$ are non-primes $p$ that happen to have $a^p=a$ mod $p$, but only for that particular $a$, not every $a$? Am I missing something? $\endgroup$
    – YiFan
    Feb 15, 2019 at 11:55
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    $\begingroup$ Sorry, I should probably have referenced Carmichael numbers (which are Fermat pseudoprimes to all bases) instead. $\endgroup$
    – hmakholm left over Monica
    Feb 15, 2019 at 12:00
  • $\begingroup$ @HenningMakholm I see, thanks for that! I didn't learn about those numbers until now. Will edit the post to reflect this! $\endgroup$
    – YiFan
    Feb 15, 2019 at 12:05
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Wilson's theorem:

$p\in \mathbb N_{>1}$ is a prime number if and only if $$(p-1)!\equiv -1 \mod p$$

Alternatively

$p\in \mathbb N_{>1}$ is a prime number if and only if $$\pi(p)=\pi(p-1)+1$$ Where $\pi(n)$ is the prime counting function, i.e. the function counting the number of prime numbers less than or equal to some real number $n$.

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  • $\begingroup$ I totally forgot about Wilson's theorem:) It is a nice definition. $\endgroup$
    – DaBler
    Feb 15, 2019 at 12:19
  • $\begingroup$ I also think so ;) $\endgroup$
    – Dr. Mathva
    Feb 15, 2019 at 12:20
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See here : http://mathworld.wolfram.com/PrimeDiophantineEquations.html for a theoretical very powerful, but in practice unfortunately not very useful method to decide whether a given number is prime. $k+2$ is prime if and only if the given diophantine equation system is solvable with the corresponding $k$.

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