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It is well known that there are infinity many primes, the integer $n=1 $ was a prime number and had been deleted from the list of primes because $1$ has one divisor which is $1$ it self, I would like to ask the similar question which states:

Question:

Why the integer $n=2$ is considerable to be a prime number however is even integer and is not similar to others primes which are odd?

Note: I would like to know a rigourous proof show that because the definition of prime number is not really enough to me with $n=2$ is different from other primes.

Thank you for any help

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closed as unclear what you're asking by Adam Hughes, Leucippus, Alexis Olson, JMP, user91500 Oct 15 '16 at 6:31

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ It helps if you start with a definition. Something which fits with the fundamental theorem of arithmetic might be good too. $\endgroup$ – Henry Oct 14 '16 at 20:59
  • $\begingroup$ somewhat related $\endgroup$ – Lionel Ricci Oct 14 '16 at 21:00
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    $\begingroup$ Even just means multiple of $2$. $2$ is the only prime that is a multiple of $2$. $3$ is the only prime that is a multiple of $3$. $5$ is the only prime that is a multiple of $5$. They’re all the same in this respect: there is nothing special about $2$. $\endgroup$ – Brian M. Scott Oct 14 '16 at 21:01
  • $\begingroup$ ok , why 1 is deleted from primes list however is odd and satisfy the prime property ? Is nothing here about odd or even .... ? $\endgroup$ – zeraoulia rafik Oct 14 '16 at 21:08
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    $\begingroup$ Because a ˆrime must have exactly two divisors, and $1$ has only one. Other than that, if $1$ were considered prime, we wouldn't have the unique factorisation theorem. $\endgroup$ – Bernard Oct 14 '16 at 21:33
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For a number to be "even" just means that it is divisible by $2$. In much the same way, I could declare that a number is "threeven" if it is divisble by $3$. Then $3$ is the only threeven prime. Do you object to $3$ being prime? Evenness is only special in the fact that it happens to be an important property of numbers so often.

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Primes have the special property that, supposing $p$ is prime, and $p$ divides a positive integer $n$, then no matter how we choose to factorise $n=ab$ into positive integers $a$ and $b$, then $p$ must also divide at least one of $a$ or $b$. This holds equally well for the number $2$ as it does for any other odd prime.

Example: $p=3$, and $n=24$ say. We can write $24=1\cdot 24, 2\cdot 12, 3\cdot 8, 4\cdot 6$, and then reversing the order. You can see the claimed property is true.

Example: consider what happens with $4$, and a number it divides like $12$, then we can write $12=1\cdot 12, 2\cdot 6, 3\cdot 4$ and the reverse order, but we immediately see the problem $12=2\cdot 6$, so four cannot be prime, which we knew, but just as an example that $4$ doesn't have the property.

$1$ is also special in that it also divides both factors in the factorisation, which in generalisations is called a 'unit' and so should be excluded from the definition of a prime.

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