A student claims that every prime greater than 3 is a term in the arithmetic sequence who nth term is 6n+1 or in the arithmetic sequence whose nth term is 6n-1. Is this true? If so why?
Look at those two arithmetic sequences:
The arithmetic sequence whose nth term is 6n+1 is 7,13,19,25,31,37,...
The arithmetic sequence whose nth term is 6n-1 is 5,11,17,23,29,35,...
One of the other of those two progressions contains every odd number
greater than 3, which is not a multiple of 3. And since every prime
number greater than 3 is odd, and is not a multiple of 3, it must be
true for every prime number greater than 3.
Quod Erat Demonstrandum
Edwin