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This is a bit like the projects searching for primes of the form k * 2n 1 except that k can only be 3, thereby boosting the speed of the process.

I am not sure this is true. k=3 is a real heavy number  dense but primeful, hopefully...
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Last “positive” result: 3 * 2^1232255 1 is prime (found on August 30th, 2004)

Wow, nearly a year