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So the other day I was in an interview for a PhD program, one of the professors asked me to give the definition of a prime number. So I gave him the following usual definition-

Any positive number $p>1$ is called a prime number if the only divisors of $p$ are $1$ and $p$ itself.

Then he asked me what are the other definitions of a prime number? To which I had no answer as I don't know any other definition. Then he said the following-

A natural number $n > 1$ is a prime number if and only if $(n-1)!=-1 (\text{mod} ~n).$

Which is basically the statement of Wilson's theorem.

My question is, can we use these kind of results//theorems as definitions of prime numbers?

If I say, "for any natural number $n>1$, if $\phi(n)=n-1,$ then $n$ is a prime, where $\phi$ is Euler's Totient function." would also qualifies to be a definition of prime number?

If so what are the other definitions of prime numbers?

Thanks for any valuable input.

Edit: So from the comments I got several examples, thanks for that, I got the idea. So now I want an answer to the question, can we use these kind of results//theorems as definitions of prime numbers?

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    $\begingroup$ If we allow Wilson's theorem to be considered a definition of a prime number - instead of viewing it as a corollary of primality by what one could call "the" definition (which you gave) - then it feels like any of the foolproof primality tests would warrant a valid definition of prime. $\endgroup$ – Eevee Trainer Mar 12 at 2:13
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    $\begingroup$ A prime number could be defined as the generators of the prime ideals of $\mathbb_{Z}$. $\endgroup$ – Chip Hurst Mar 12 at 2:15
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    $\begingroup$ Or the positive integers for which $\Bbb{Z}/p\Bbb{Z}$ is a field. $\endgroup$ – Servaes Mar 12 at 2:16
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    $\begingroup$ One of the most important ways you can define a prime integer $p$ is as an integer for which the implication $p\mid ab$ implies $p\mid a$ or $p\mid b$ holds for all integers $a,b$. $\endgroup$ – JMoravitz Mar 12 at 2:21
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    $\begingroup$ The points of discontinuity of $\pi(x) = R(x) - \sum_\rho R(x^\rho) -1/\ln(x) - \tan^{-1}(\pi/\ln x)/\pi$, where $R$ is the Riemann R function and $\rho$ is a nontrivial zero of the zeta function. $\endgroup$ – eyeballfrog Mar 12 at 2:26
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Just like Wilson's theorem (a positive integer is a prime if and only if ....) any equivalent statement is handy when we want to show a number is prime. If we encountered a number in a situation where congruence condition of its factorial is available then it is good to use that for testing primality.

Logically any of the "if-and-only-if theorems" can be taken as a definition.

But human beings who discover theorems are not logical machines. While learning a subject, a formulation that uses less preliminary concepts is helpful; So the theorem "$n$ is a prime iff $Z/nZ$ is a field", is not suitable as a definition.

Accepted textbooks use that formulation of definition which makes it easy to digest and understand the subject, even if the subject did not evolve that way historically!

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Take the statements of any of the deterministic pirmality tests. Each of their converse gives a new definition of primes.

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I'm going to elaborate on the comment that a number $p$ is prime if $p \mid ab$ means either $p \mid a$ or $p \mid b$.

Consider for example $p = 14$. That's not actually prime, but indulge me for a minute. Check that $14 \mid 112$ and $112 = 7 \times 16$. However, $14 \nmid 7$ and $14 \nmid 16$ either. Therefore $14$ is not prime. However, $2$ and $7$ are prime, since for any choice of $a$ and $b$ such that $ab = 112$, you will see that either $a$ or $b$, maybe both, are divisible by $2$ and/or by $7$.

The definition you gave is equivalent to saying if $ab = p$, then either $a$ or $b$ is a unit (like $-1$). And that definition is fine until you venture out to domains of numbers other than $\mathbb{Z}$. For example, are the numbers $3 \pm 2 \sqrt{10}$ prime? Yes, they are. How about $\sqrt{10}$? It satisfies the $ab = p$ definition but not the $p \mid ab$ definition.

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