Whoa! The primes are acting weird!
What percent of primes end in a 7? I mean when you write them out in base ten.
Well, if you look at the first hundred million primes, the answer is 25.000401%. That looks suspiciously close to 1/4. And that makes sense, because there are just 4 digits that a prime can end in, unless it's really small: 1, 3, 7 and 9.
So, you might think the endings of prime numbers are random, or very close to it. But 3 days ago two mathematicians shocked the world with a paper that asked some other questions, like this:If you have a prime that ends in a 7, what's the probability that the next prime ends in a 7?
I would still expect the answer to be close to 25%. But these mathematicians, Robert Oliver and Kannan Soundarajan, actually looked!
And they found that among the first hundred million primes, the answer is just 17.757%. That's way
So if a prime ends in a 7, it seems to somehow tell the next prime "I rather you wouldn't end in a 7. I just did that."
Pardon my French, but this is fucking weird. And I'm not just saying this because I don't know enough number theory. Ken Ono is a real expert on number theory. And when he learned about this, he said:“I was floored. I thought, ‘For sure, your program’s not working.’ "
Needless to say, it's not magic. There will be an explanation. In fact, Oliver and Soundarajan have conjectured a formula that says exactly how much of a discrepancy to expect - and they've checked it, and it seems to work. It works in every base, not just base ten. But we still need a proof that it really works.
By the way, their formula says the discrepancy gets smaller and smaller when we look at more and more primes. If we look at primes less than N, the discrepancy is on the order of log(log(N))/log(N). This goes to zero when N goes to infinity. But this discrepancy is huge compared to the discrepancy for the simpler question, "what percentage of primes ends in a given digit?" For that, the discrepancy - the difference between reality and what you'd expect if primes were random - is on the order of 1/(log(N) sqrt(N)).
Of course, what's really surprising is not this correlation between the last digits of consecutive primes, but that number theorists hadn't thought to look for it until now!
For more, read the article below, and of course the actual paper:
• Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, http://arxiv.org/abs/1603.03720