Question

1. Show that there exists a prime number p such that p+4 and p+6 are also prime. [Hint: Primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, ...]

Answer 1

There exists three prime numbers such that p , p+4 and p+6 are also prime . These are 3,5 and 7.

Explanation:

In order to see this, we first note that the only prime p which is congruent to 0 (mod 3) is 3 itself. We can now proceed by considering several cases. If p ≡ 0 (mod 3) then as we noted above the only way p can be prime is if p = 3. In this case one can easily verify that p + 2 = 5 and p + 4 = 7 are also prime. If p ≡ 1 (mod 3) then p + 2 ≡ 0 (mod 3) and therefore in order to be prime p + 2 would have to be equal to 3. However, this would imply that p = 1 and 1 is not a prime number. Therefore, there are no prime triplets with p ≡ 1 (mod 3). If p ≡ 2 (mod 3) then p + 4 ≡ 6 ≡ 0 (mod 3). Thus by above in order for p + 4 to be prime it would have to equal 3, which is impossible. Because we know that p must be congruent to 0, 1, or 2 (mod 3) we have now exhausted all possibilities and shown that the only prime triplet is (3, 5, 7).