Great piece from Slate.
Why do we know about so many large Mersenne primes and so few large non-Mersenne ones? It’s not because large Mersenne primes are particularly common, and it’s not a spectacular coincidence. That brings us back to the road and the street lamp. There are several different versions of the story. A man, perhaps he’s drunk, is on his hands and knees underneath a streetlight. A kind passerby, perhaps a police officer, stops to ask what he’s doing. “I’m looking for my keys,” the man replies. “Did you lose them over here?” the officer asks. “No, I lost them down the street,” the man says, “but the light is better here.”
We keep finding large Mersenne primes because the light is better there.
First, we know that only a few Mersenne numbers are even candidates for being Mersenne primes. The exponent n in 2n-1 needs to be prime, so we don’t need to bother to check 26-1, for example.* There are a few other technical conditions that make certain prime exponents more enticing to try. Finally, there’s a particular test of primeness—the Lucas–Lehmer test—that can only be used on Mersenne numbers.
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