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13 February, 11:24

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

Answers (2)
  1. 13 February, 11:43
    That is true. 2 and 3 are the only consecutive prime numbers. it equals 5 which is also a prime number.
  2. 13 February, 13:46
    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


    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.
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