- Among these three possibilities, only 3k+1 and 3k+2 can be primes and 3 is a prime.
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.
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 , ,
They're all prime numbers.
And it's the only possibility for p.