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