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  • 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.

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We found a new pattern in the primes! One that we didn't know was there.

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質數的最後一位數字 - Numberphile(數字愛好者) (The Last Digit of Prime Numbers - Numberphile)

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    蔡元皓 發佈於 2021 年 01 月 14 日
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