If the numbers p, p+2 and p+4 are primes, find p.

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- Mar 20th 2009, 12:04 PMApprentice123Prime numbersIf the numbers p, p+2 and p+4 are primes, find p.
- Mar 20th 2009, 12:16 PMMoo
Hello,

- Any integer can be written in the form 3k,3k+1 or 3k+2, for some integer $\displaystyle k \geq 0$

- Among these three possibilities, only 3k+1 and 3k+2 can be primes and 3 is a prime.

Now if $\displaystyle p=3k+1$

$\displaystyle {\color{red}p+2=3k+3=3(k+1)}$

$\displaystyle p+4=3k+4$

**p+2**is prime if and only if k+1=1, that is k=0. Otherwise, it is a multiple of 3 and hence is not prime.

But p=3x0+1=1 is not a prime.

Hence no prime number in the form 3k+1 satisfy the condition.

Now if $\displaystyle p=3k+2$

$\displaystyle p+2=3k+4$

$\displaystyle {\color{red}p+4=3k+6=3(k+2)}$

**p+4**is prime if and only if k+2=1, that is k=-1.

This gives p=-3, p+2=-2, p+4=1.

These are obviously not prime numbers. (they have to be positive).

Now if $\displaystyle p=3$, $\displaystyle p+2=5$, $\displaystyle p+4=7$

They're all prime numbers.

And it's the only possibility for p. - Mar 20th 2009, 02:28 PMApprentice123
Thank you very much, the exercise was not easy.