Let us call a sequence (a_n) primitive if it has the property that gcd(a_i,a_n) = 1, for all 'i' less than n.

Question: State whether the following statements are true or false.

The sequence of mersenne numbers M_n = 2^n -1 is a primitive sequence?

The sequence of mersenne numbers M_p = 2^p -1 (here p shall only take on prime numbers) is a primitive sequence?

The sequence of Fermat numbers 2^(2^n) +1 is a primitive sequence?

(Hard question) The sequence of numbers defined by a_n = 2*3*5....*p_n -1 (p_n is the nth prime number) is a primitive sequence.

^