it is well known that the integer $2$ is a prime number as it satsfying the definition of prime number , Another remark is that the presentation of the integer 2 is different from others primes , for example (it is the product of even integer times $1$ ($2=2\times 1$ ) but others primes are the products of odd integers times 1 , Then my question here is : Why $2$ is a prime number however it has differents representation than other odd primes ?

Note: Probably the unit integer which it is 1 excluded from the list of primes as a reason it's representation as the product of the number times it self ($1=1\times 1$)which it is different from other primes representation


closed as unclear what you're asking by Jyrki Lahtonen Feb 26 '18 at 23:07

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    $\begingroup$ You're saying that "2 has a different representation" and the "difference" is that 2 is "even." But this statement has no actual content, because the word "even" just means "divisible by 2." $\endgroup$ – Joshua Ruiter Feb 26 '18 at 23:04
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    $\begingroup$ 3 also has a unique representation: it is the only prime that is the product of a multiple of 3 times 1. Likewise 17 is the only prime that is the product a multiple of 17 times 1. If you think those are silly examples, bear in mind that "an even integer" just means "a multiple of 2." $\endgroup$ – David K Feb 26 '18 at 23:05
  • $\begingroup$ The representation i meant is in the side of parity $\endgroup$ – zeraoulia rafik Feb 26 '18 at 23:06
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    $\begingroup$ "Parity" just means "multiple of 2" vs. "not multiple of 2." $\endgroup$ – David K Feb 26 '18 at 23:07
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    $\begingroup$ Anyway, numbers are designated as primes because it is useful to call them primes. Putting 1 in the set of primes would make a lot of useful facts about "primes" a lot more complicated; removing 2 from the set of primes would also mess up a lot of facts about "primes". $\endgroup$ – David K Feb 26 '18 at 23:12

Note that

  • $1$ is to be excluded since every integer number is multiple of $1$
  • $2$ is included since it is the "first even" number; it is true that the others primes are odd (i.e. not multiple of 2) but this holds because the key property is that primes are not multiple of the previuos numbers other than 1.
  • $\begingroup$ I don't know why the question is closed however it is clear at all what i asked ? $\endgroup$ – zeraoulia rafik Feb 26 '18 at 23:12
  • $\begingroup$ What do you mean exactly with "different representation"? $\endgroup$ – gimusi Feb 26 '18 at 23:14
  • $\begingroup$ representation in the contex of product for example : 2 presented as even integer times 1, but all other primes have the same property in the contex of product representation: odd times 1 $\endgroup$ – zeraoulia rafik Feb 26 '18 at 23:18
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    $\begingroup$ It’s best to accept first the definition of composite number: a number that has factors other than 1 and itself. No parity oddity to consider in this definition. Prime numbers are the numbers that are not composite. $\endgroup$ – ir7 Feb 26 '18 at 23:39
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    $\begingroup$ @zeraouliarafik did all the comments given help you? what else is not clear to you? $\endgroup$ – gimusi Feb 26 '18 at 23:53

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