The numbers 3n−1 are never prime (if n ≥ 2), since they are always even. However, it sometimes happens that (3n−1)/2 is prime. For example, (33−1)/2 = 13 is prime.
(a) Find another prime of the form (3n − 1)/2.
(b) If n is even, show that (3n − 1)/2 is always divisible by 4, so it can never be prime.
(c) Use a similar argument to show that if n is a multiple of 5 then (3n − 1)/2 is never a prime.
(d) Do you think that there are infinitely many primes of the form (3n − 1)/2?
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