A student claims that every prime greater than 3 is a term in the arithmetic sequence whose nth term is 6n + 1 or in the arithmetic sequence whose nth term is 6n - 1. Is this true? If so, why?
A student claims that every prime greater than 3 is a term in the arithmetic sequence whose nth term is 6n + 1 or in the arithmetic sequence whose nth term is 6n - 1. Is this true? If so, why?
yes, the claim is still true for n = 4
the next prime is 23 which is of the form 6n - 1 with n = 4. now you might say, "but 6n + 1 = 25 when n is 4, which is not prime, so the claim is false!" remember, we have "or" here. if one form does not give a prime, it is fine. as long as all the primes can be written as one form or the other.
i don't think trial and error way is the way to go here.
to flesh out what i hinted at last time...
EVERY integer can be written as one of the following forms for some integer n: 6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4, or 6n + 5. the claim is, if our integer is a prime greater than 3, it is of the form 6n + 1 or 6n + 5 (which as i said, is equivalent to 6n - 1, still waiting for you to tell me why).
now, assume the claim is false, then it means we can have a prime of one of the other forms, but not of the two we were considering. start with 6n + 2, note that 6n + 2 = 2(3n + 1). but that means any integer of that form is even, that is, it is a multiple of 2 and hence cannot be prime. what about the other cases?