For starters, I am not a mathematician, but I was fascinated by prime numbers for many years. So today, I came up with the definition which I am extremely curious about, because I believe it poses a mathematical problem.
Prime number is a positive integer that has exactly two positive integer divisors.
Here, I assume that if p is a positive integer, then 1 and p are among its divisors.
So the question, is there a positive integer number which is not prime and which has exactly two positive integer divisors?
In case it is already known definition, I would be thankful if you pointed me to it.
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$\begingroup$The number $1$ has exactly one positive integer divisor, namely $1$.
If $a$ is prime, then it has exactly two positive integer divisors, namely $1$ and $a$.
If $a$ is not $1$ and not prime, then it has at least three positive integer divisors: $1$ and $a$ and at least one other.
The definition in the OP:
Prime number is a positive integer that has exactly two positive integer divisors.
tells us the meaning of the word prime for this purpose.
Why is the definition stated like this? Long ago, the definition may have been: "A number $a$ not divisible by any positive integer except $1$ and $a$." This definition would include $1$ as a prime. When mathematicians found it more useful not to include $1$ among the primes, they came up with the defintion in the OP.
$\endgroup$ 0 $\begingroup$A natural number always has the two divisors one and itself. So it saying "exactly two divisors" and "only the divisors one and itself" are equivalent statements. The first is more compact.
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