Question

True or False: the sum of any two consecutive prime numbers is also prime?

Answer 1

Let's consider two prime numbers where each is larger than 2

Say the primes 7 and 11. Adding them gets us 7+11 = 18. This counter example disproves the initial claim since 18 = 9*2 = 6*3 making 18 composite (not prime)

In general, if we let p and q be two primes such that q > p > 2 and q is the next prime after p, then p and q are both odd. If any of them were even then they wouldn't be prime (2 would be a factor)

Adding any two odd numbers together leads to an even number

Proof:

p = 2k+1 where k is some integer

q = 2m+1 where m is some integer

p+q = 2k+1+2m+1 = 2(k+m) + 2 = 2(k+m+1) which is in the form of an even number

That proof above shows us that adding any prime larger than 2 to its next prime up leads to an even number. This further shows us that the claim is false overall. It is only true if you restrict yourself to the primes 2 and 3, which add to 5. Otherwise, the claim is false. 

Answer 2

That is true. 2 and 3 are the only consecutive prime numbers. it equals 5 which is also a prime number.