Originally published at: "Sexy primes" are prime numbers that differ by six | Boing Boing
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What about 1 and 7? Why is it skipped?
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Nowadays, one is not considered to be a prime number since primes are defined as only being divisible by one and themselves.
This hasn’t always been the case - in the 18th and 19th Centuries, most mathematicians considered one to be prime; but the Greeks who discovered primes didn’t even consider one to be a number.
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Primes that differ by 6 are no more silly than primes that differ by 2, and the twin prime conjecture is one of the most famous outstanding conjectures in mathematics–a proof would be huge (even a result that makes a tiny bit of progress towards a proof is a major chievement, so Yitan Zhang’s result proving there are an infinite number of prime pairs differing by some number that is less than 70 million was shocking to mathematicians, and actually made front page news in 2013.
Plus there are lots of reasons why working mod 6 is interesting, 6 being the first product of distinct primes itself. All odd primes are either 1 more than a multiple of 6, or one less than a multiple of 6.
So when you say it’s a silly recreational math problem, you just mean it has a silly name. I just thought I’d clear that up!
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You can’t have primes separated by an odd number (unless one of those primes is 2) but what about even numbers? Are there pairs for all even numbers? Are any such pairing provably finite, would they all be like the twin primes conjecture - likely infinite, but not proven? Or maybe some or provenly infinite?
The fact that it raises questions like these certainly makes them seem less “silly”.
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While I agree with 1 not being a prime, that definition always bothered me because 1 also can only be divided by 1 and itself. It doesn’t specify that they have to be two different numbers.
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If 1 were considered prime, then there would not be a unique prime factorization for any integer. That is the only reason it isn’t considered prime.
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Thanks. I KNEW there was a reason for not regarding 1 as prime, but damned if I could remember what it was.
Also, the “sexy” joke probably makes more sense to New Zealanders.
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The other day my six-year-old ask if all the numbers had been found yet. He’d just counted to fifty by himself for the first time, so I suspect he was wondering how high one could count. But what a wonderful question - and I had to tell him no, some numbers haven’t been found yet. We then had a conversation about some numbers being special - which is pretty hard to frame when you’ve talking to someone who doesn’t yet know about division, but it went pretty well.
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