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I was considering the possibility of writing primes (except 2 and 3) in the form 2n + 3 for some positive integer n. Is it always possible ?

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    $\begingroup$ This just asks whether every prime $p>3$ is odd, i.e., of the form $2m+1$ with $m>1$ - which is clearly true. $\endgroup$ Jan 13, 2017 at 19:26

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Yes, it is always possible. Indeed a prime number $p>3$ is odd and therefore can be written $p=2m+1$ with $m > 1$ (since $p>3$). Then $p=2(m-1+1)+1=2(m-1)+3=2n+3$ with $n=m-1$ a positive integer.

Note that this result is very general and does not help to distinguish prime numbers since it is true for every odd number greater than 3.

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Every odd number $p$ greater than 4 can be written in the form $2n + 3$; just choose $n = \frac{p-3}{2}$. So this statement is certainly true for all primes other than 2 and 3.

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