a. If n is even, show that (3^n-1)2 is always divisible by 4, so it.pdf

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a. If n is even, show that (3^n-1)/2 is always divisible by 4, so it can never be prime. b. Use a similar argument to show that if n is a multiple of 5 them (3^n-)/2 is never a prime. c. Are there infinitely many primes of the form (3^n-1)/2? Solution a) n is even, let n=2k (3^(2k)-1)/2 = (9^k-1)/2 = [(8+1)^k-1]/2 = (8p+1-1)/2 = 4p So it is divisble by and is not a prime. b) let, n = 5k (3^(5k)-1)/2 = (243^k - 1)/2 = ((242+1)^k- 1)/2 = 242p/2 = 121p = 11^2p Hence it is divisible by 11 and is not a prime. c) Yes..

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