We found a new pattern in the primes! One
that we didn't know was there. Primes don't like to repeat their last digits.
It's really, really strange. A prime could end with a one, three, a seven, or a nine. I mean the exceptions are two
itself, which obviously end in a two and five itself. But apart from that, they end with a one, a three, a seven, or a nine, right. And because we hope that primes are random, or they feel like they kind of appear randomly, that each one of those should be equally likely and something
strange has happened, so some mathematicians in Stanford have looked at this and they've
looked at consecutive primes and they've noticed things like if a prime ends in
a nine, it's actually more likely to, for the next prime to end in a one, rather than a nine or a three or a seven. So you would expect all of those to equally likely
to turn up next but when they did it and they looked at the
first hundred million primes and when they did that, if you had a prime ending in
a nine the chance the next prime ended in a nine was eighteen percent, which is not a
quarter, and the chance that ended in a one was greater than a quarter, it was thirty-two percent. It shouldn't be! That shouldn't be how it is, but that's what they found out. Let's find out why this is. So
if the primes are random, then these prime endings here: one, three, seven, and nine, they should each appear about a quarter of the time, and if we had the consecutive primes and we look at
their prime endings we would have sixteen options: You have a one and a one, so that would be a prime ending in a one followed by a prime ending in a one. Or it could be a one followed by a three, a one followed by seven, a one followed by nine These are the options. Ok, there are sixteen options when you're
looking at two consecutive primes so if it was random they should all appear
one-sixteenth of the time equally likely of the time. That's not they've found at all. In particular, these diagonal entries here - so the one, one; three, three; seven, seven; nine, nine - are least likely to turn up. So, the primes aren't
repeating themselves and there's a few explanations for why this could be the
case which I'm going to dismiss. I'm going to give you a few explanations which aren't the reason why this is happening. So one might be: "Okay if you have a
prime ending with a nine, then well you have to go through all the other numbers
before you get to another prime ending with a nine." Or, you have to go through the numbers
that end with a one, and a three, and a seven before you get to another number ending with a nine, so. [Brady Haran] It's further away! [James Grime] It's further away, right. And unfortunately, that isn't enough to explain the bias that we found. if that's the case then we're
looking for prime gaps less than ten. Right? That means you have the next one in less than ten so you're going for a prime ending in a one next, or the prime ending with a three next. and prime gaps less than ten are not that many and so that's not enough to explain how biased this is. The bias is bigger than that, so
that's no good. Another explanation might be: I
said that these prime endings one, three, seven and nine if this was a random thing they should be
appearing a quarter of the time each. Maybe the explanation is that that's not a true thing maybe the bias comes from the primes themselves and that doesn't work
out either there is a slight bias in the primes and it's something we know
about. It's called Chebyshev's bias. It says that primes ending in three and
seven are slightly more likely. That is something we know about, we know why
that happens, and it's such a slight thing. It is not enough. So there is this known
bias in prime number endings, so under that assumption, the pair three-three and seven-seven should turn up more often, right? And the complete opposite is true. In fact, three-three and seven-seven are the least likely that are turning up. which is the exact opposite of what it
should be, and nine-one are actually the most common pair! which is not what it should be at all. Ok, maybe, oh maybe it's just a base 10
thing, you know. Base ten, you know, who cares about base ten, right? If it was a fundamental property
of primes, it would happen in any base. And it does. That's what they found. So they checked it in other bases and they found the bias is still there, so it
appears to be a fundamental property of the primes. For example, if we do it in
base three. It will only end with a one or two unless it's three itself, right, we can ignore that one. You got two endings they should turn about fifty percent of the time each, which is about
right, again Chebyshev's bias says just a little, slight bias toward the two, but its tiny.
It's pretty much fifty-fifty, it's pretty much a coin toss. In fact that's what inspired
this investigation, and the guys who did this investigation were thinking
about coin tosses and said "Well primes in base three are like a coin toss. Let's see if that's the same thing because that's a random event." And then they found this completely different thing, this skew that primes don't like to repeat themselves. And that's not something that would happen in coin tosses. So if we did it in base three, we would have four prime endings, wouldn't we? We would have one-one, one-two, two-one, and two-two. And again they
found the same thing. These ones with the repetition are the least likely to
occur. They looked at the first million primes. And if you look at first million primes, then
they should all be equally likely to turn up a quarter of the time: Two-hundred and fifty thousand. But no, they didn't get that, so these
ones, with the repetition, were less than two hundred and fifty thousand. The ones without a repetition were more than two hundred and fifty thousand. So the mathematicians who have been investigating this have tried to come up with an explanation for this. And their explanation relies on a
conjecture that goes back a hundred years. It's a conjecture by Hardy and Littlewood and they had a conjecture about the density of primes: how many primes you can find in patterns. So you can consider all kinds of patterns Like twin primes, that's a pattern, or cousin primes, which have gaps or four, or sexy primes, who have gaps of 6. So they had this conjecture about how many of these
you should find, and the conjecture has not been proven There's a lot of evidence that supports
that it's true, so if you look at the numerical evidence, it appears to be true but it hasn't been proven. So the mathematicians looking at this pattern used a modified version of that conjecture and they came up with a formula that they think might
explain this idea. So that formula was the proportion of this pattern - let's say you've got prime endings a and b, so if we're doing it in base ten, this could be one-one or three-seven, or nine-one. So we're looking at the proportion of these endings and they come up with a formula. The formula was: one over the number of allowed endings, so this is like one sixteenth from what I've been doing in base ten right? So when it's equally likely, these are the allowed endings there. Sixteen of them. So the proportion is one over the
allowed endings multiplied by a thing. Right, and what is this thing? That thing depends on if that pattern repeats, if you've got a equals b in that formula. If a equals b, that will affect what that
thing is. I'll show you what it looks like in base three. It's kind of ok in base three. We're looking at proportions of these endings. If they're the same like this: a, a. So that would, in base three, the one-one endings and the two-two endings. The proportion is. If you are doing it when they're not equal, so if it was the one-two endings or the two-one, the proportion of a b, and I'm saying a is not the same as b Plus. So this formula they've got is still a conjectured formula because it is based on this Hardy-Littlewood formula which is still a conjectured formula. But it fits the evidence. Once we start going off to infinity, this bias becomes less and less important. This is a bias that is hanging around so in the great infinity of numbers, this bias is evening out, but even up to a trillion there is still a noticeable bias there. Audio books are a great way to pass time if, for example, you spend a lot of time commuting, and with over two hundred and fifty thousand titles in its collection, audible.com is the place to find
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I think it's important to note (he sort of says this at the end of the video) that the asymptotic density of these pairs IS what you would expect- 1/16 in the base 10 case. So the bias is a factor which tends to zero as you go higher.
For the Chebyshev Bias, check out this article on the Prime Race. Essentially, even though the primes are equally distributed about last digits or, in general, remainders, if we look up to some point we'll generally find a bias. If we look at the remainders of primes after dividing by 4, then there will usually be more with remainder 3 than remainder 1 (though they even out towards infinity). If the primes were independently distributed, then we'd expect to see pairs of consecutive primes with (3,3) more than (3,1), (1,3) or (1,1), simply because there are more primes with remainder 3 to begin with. But the numerics suggest that (3,1) and (1,3) are far more common so things are not as independent as we thought.
Perhaps I'm being stupid, but the first idea he dismissed wasn't fully dismissed. It would matter most for primes less than 10 digits away, half as much for primes less than 20 digits away, etc. The bias from that wouldn't go away after completely after the first 10 numbers.