The Last Digit of Prime Numbers - Numberphile
video description
Hopefully all that yelling garnered your attention. About 5 months ago I explained the screw-up by these two Stanford profs. And it's been about 1 1/2 years since I explained it on the Quantus
Mag website. In the Quantus article the profs gave -a simple example- problem to demonstrate how misleading a math result could seem. And-amusingly enough-they did that problem wrong too!
Well I've been waiting for a nice pat-on-the-back. but fruitlessly so. And, admittedly, I only used -40 words to explain the key point.
So let's revisit my former comment & I will explicate more thoroughly. I assure you the reasoning is not complex. Give me a chance!
The supposed anomaly is totally explained by 2 basic things.
(1) The first point is mentioned in the video, but I will reiterate. Suppose you have a large prime # ending in 1: let's call it 01, letting this represent a large prime number ending in the two digits 01. Here are the next possible primes: 03, 07, 09, 11. So it is partially explained by the fact that 03, 07, 09 get a shot at it before the next # ending in 1.
Now here is how I explained point #2:
-(2) Since 01 is prime it is not divisible by 3. But then there is a 50% chance that 11 is! And while there is a 50% chance that 03 & 09 are; clearly 07 is not divisible by 3. Get it! -
Let's explicate that in a little more detail:
Recall that 01 is a large prime number ending in 01; so it might be like 537. 701, the -. - representing missing digits. Since it is prime, of course it's not divisible by 3. The next number 02 might be divisible by 3 as might 03. But 04 cannot be as it is 3 more than 01 which we know is not. There are 3 possibilities for a number n: it can be 0, 1, or 2 modulo 3.
Now do you see why there is a 50% chance that 11 is divisible by 3? As 01 is not, 04, 07, 10, 13, .are also not. If 02 were, then 05, 08, 11, 14 would also be; the only other possibility is that 03, 06, 09, 12, 15. were divisible by 3.
I hope statement (2) is now clear. I went on to note:
- Larger primes than 3 also have an effect but it is reduced as they get larger. And perhaps I should point out that when you calculate this you should not just put in 50% probabilities but to be accurate figure out the probabilities of each branch. Thus one branch has 03 09 21. divisible by 3; the other has 11, 17, 23, . divisible by 3. Hindsight being 20-20 we may only note that they should have had someone check this out before going public or publishing. And most especially in Probability which always is especially tricky. -
Thus, for example, we would next consider the prime # 7. 01, 08, 15, 22, .are not divisible by 7. For #s not on this sequence there would be a 5/6 probability of its being not divisible by 7. Note, however, to be accurate, we would need to figure out the probability going down each branch: assuming, in one case, for example, that 02 was divisible by 7; & so on.
Hope this clarifies things.
Let's see if anyone's paying attention!
Date: 2022-04-08
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Comments and reviews: 9
Des
I'm a little late but the thing about looking at gaps less than ten seems off to me. I mean, it is apparent the bias is larger than this would cause, but I believe this is not exactly negligible.
Say that primes really are -randomly- distributed, any number with allowed ending has a chance to be prime and this chance depends on how big numbers we're dealing with. Considering the chance for a number to not be a prime increases logarithmically as the numbers grow, for large enough numbers the difference between the chance for n and say n+20 to be a prime is negligible. Let's call it p.
The chance next allowed numbers are a prime would be:
p x (1-p)-0, p x (1-p)-1, p x (1-p)-2, p x (1-p)-3,
p x (1-p)-4, p x (1-p)-5, p x (1-p)-6, p x (1-p)-7.
In other words, not only gap 10 is less likely than any of the 2-8 ones, but also gap 20 is less likely than any of the 12-18 ones, and so on. Actually, in columns, we got geometric series to work out, all with a quotient of (1-p)-4.
a = (p x (1-p)-0) / (1 - (1-p)-4)
b = (p x (1-p)-1) / (1 - (1-p)-4)
c = (p x (1-p)-2) / (1 - (1-p)-4)
d = (p x (1-p)-3) / (1 - (1-p)-4). chance for same ending
Let's take the chance for d over the chance for a:
(p x (1-p)-3) / (1 - (1-p)-4) / (p x (1-p)-0) / (1 - (1-p)-4) = (p x (1-p)-3) / (p x (1-p)-0) = (1-p)-3
If p is say 5% (And here we're dealing with numbers of the size -e-20 = 4. 85e8, the chance of same ending happening is still 0. 857375 times the chance of the -next- ending happening (so like 3 after 1 or 7 after 3. And p doesn't shrink any fast, actually it stays above 4% for numbers less than -e-25 = 7. 20e10.
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I'm a little late but the thing about looking at gaps less than ten seems off to me. I mean, it is apparent the bias is larger than this would cause, but I believe this is not exactly negligible.
Say that primes really are -randomly- distributed, any number with allowed ending has a chance to be prime and this chance depends on how big numbers we're dealing with. Considering the chance for a number to not be a prime increases logarithmically as the numbers grow, for large enough numbers the difference between the chance for n and say n+20 to be a prime is negligible. Let's call it p.
The chance next allowed numbers are a prime would be:
p x (1-p)-0, p x (1-p)-1, p x (1-p)-2, p x (1-p)-3,
p x (1-p)-4, p x (1-p)-5, p x (1-p)-6, p x (1-p)-7.
In other words, not only gap 10 is less likely than any of the 2-8 ones, but also gap 20 is less likely than any of the 12-18 ones, and so on. Actually, in columns, we got geometric series to work out, all with a quotient of (1-p)-4.
a = (p x (1-p)-0) / (1 - (1-p)-4)
b = (p x (1-p)-1) / (1 - (1-p)-4)
c = (p x (1-p)-2) / (1 - (1-p)-4)
d = (p x (1-p)-3) / (1 - (1-p)-4). chance for same ending
Let's take the chance for d over the chance for a:
(p x (1-p)-3) / (1 - (1-p)-4) / (p x (1-p)-0) / (1 - (1-p)-4) = (p x (1-p)-3) / (p x (1-p)-0) = (1-p)-3
If p is say 5% (And here we're dealing with numbers of the size -e-20 = 4. 85e8, the chance of same ending happening is still 0. 857375 times the chance of the -next- ending happening (so like 3 after 1 or 7 after 3. And p doesn't shrink any fast, actually it stays above 4% for numbers less than -e-25 = 7. 20e10.
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Enzo
FUN FACT: I found out that if you treat any number like a concatenation, then all the numbers resulting from a concatenation of two primes, like for example 1317 (which is a concatenation of 13 and 17) can only have a chance of being a prime number if the difference between the two primes is a multiple of 6. In the case of 13 and 17 the difference is 4 (17-13) so 1317 has no chance of being a prime. And the same is true for 1713. On the other hand, 1319 is a prime number and so is 1913. This is possible only because the difference between 19 and 13 is a multiple of 6. As a general rule: whenever you can split a number in a concatenation of two primes, the number itself has no chance of being a prime unless the difference between the two primes is a multiple of 6. There is much more about concatenations, and I believe the study of them is the only way to solve the many problems with prime numbers. When we deal with huge numbers, the only reasonable way to study them, is to treat them for what they are, long series of digits. Thus the importance of the concatenation approach to the study of prime numbers.
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FUN FACT: I found out that if you treat any number like a concatenation, then all the numbers resulting from a concatenation of two primes, like for example 1317 (which is a concatenation of 13 and 17) can only have a chance of being a prime number if the difference between the two primes is a multiple of 6. In the case of 13 and 17 the difference is 4 (17-13) so 1317 has no chance of being a prime. And the same is true for 1713. On the other hand, 1319 is a prime number and so is 1913. This is possible only because the difference between 19 and 13 is a multiple of 6. As a general rule: whenever you can split a number in a concatenation of two primes, the number itself has no chance of being a prime unless the difference between the two primes is a multiple of 6. There is much more about concatenations, and I believe the study of them is the only way to solve the many problems with prime numbers. When we deal with huge numbers, the only reasonable way to study them, is to treat them for what they are, long series of digits. Thus the importance of the concatenation approach to the study of prime numbers.
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Richard
An experiment: Set up a list of nonnegative integers in base 6. Top row: 0, 1 (each is neither prime nor composite); 2nd row: 2, 3, 4, 5, 10, 11; 3rd row 12 [=8], 13 [=9], 14 [=10, 15 [=11], 20 [=12], 21 [=13]; etc. Note that every prime after 3 falls in either column 5 or column 11 [=7]. Two facts become clear: a) All prime pairs after (3, 5) fall in columns 5 and 11, paired in the same row (e. g. 15, 21; i. e, in base ten, 11, 13. b) All primes end in either digit 5 or digit 1. So I believe that the mysterious property discussed in this video should, in base 6, simplify to this [expected] observation: The next successive prime following a prime in column 5 will be a prime found in column 11 (i. e. a prime that ends with the digit 1, more often than it will be another prime in column 5 (a prime ending in 5. Likewise, the next prime following a prime that ends in 1 will more often be a prime that ends in 5, than it will be another prime that ends in 1.
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An experiment: Set up a list of nonnegative integers in base 6. Top row: 0, 1 (each is neither prime nor composite); 2nd row: 2, 3, 4, 5, 10, 11; 3rd row 12 [=8], 13 [=9], 14 [=10, 15 [=11], 20 [=12], 21 [=13]; etc. Note that every prime after 3 falls in either column 5 or column 11 [=7]. Two facts become clear: a) All prime pairs after (3, 5) fall in columns 5 and 11, paired in the same row (e. g. 15, 21; i. e, in base ten, 11, 13. b) All primes end in either digit 5 or digit 1. So I believe that the mysterious property discussed in this video should, in base 6, simplify to this [expected] observation: The next successive prime following a prime in column 5 will be a prime found in column 11 (i. e. a prime that ends with the digit 1, more often than it will be another prime in column 5 (a prime ending in 5. Likewise, the next prime following a prime that ends in 1 will more often be a prime that ends in 5, than it will be another prime that ends in 1.
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Tad
Much ado about nothing. Noticing a bias in the first million primes is no more insightful than finding a bias in the first five primes; when both biases can be shown to be diminishing as the number of primes observed approaches infinity.
The bias is in our own human desire to extrapolate localized patterns into functions. We agree that five is a small number but fall into the trap of believing that a million is a large number. However, in the grand scheme of all primes, the group of the first million primes is no bigger sampling than a single gain of sand within all the deserts across all the planets across all the universe.
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Much ado about nothing. Noticing a bias in the first million primes is no more insightful than finding a bias in the first five primes; when both biases can be shown to be diminishing as the number of primes observed approaches infinity.
The bias is in our own human desire to extrapolate localized patterns into functions. We agree that five is a small number but fall into the trap of believing that a million is a large number. However, in the grand scheme of all primes, the group of the first million primes is no bigger sampling than a single gain of sand within all the deserts across all the planets across all the universe.
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Mara
I need someone like James to help me with a paper I have been trying to write for years. I don't have a degree because my scholarship's foundation went bankrupt, and I never even got to take a class in what I want to publish in before I had to leave. Only one person in my city has a degree in it (that I can find, I basically never had a proof writing class because of health problems (showed up to less than 5 classes and I showed up for midterms and finals to pass, and I can't even find examples in books or online of proofs using the methods I used. And cuban primes are not really known or studied like Mersenne primes -_-
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I need someone like James to help me with a paper I have been trying to write for years. I don't have a degree because my scholarship's foundation went bankrupt, and I never even got to take a class in what I want to publish in before I had to leave. Only one person in my city has a degree in it (that I can find, I basically never had a proof writing class because of health problems (showed up to less than 5 classes and I showed up for midterms and finals to pass, and I can't even find examples in books or online of proofs using the methods I used. And cuban primes are not really known or studied like Mersenne primes -_-
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Roger
This is the type of thing that fascinates me about mathematics: things that reveal fundamental properties of number, regardless of the base used. I find such things fascinating and mysterious, moreso than just about anything. It leads me to ask what is number? How does it exist? Why and how do mathematical properties such as this arise? And what is the mode of existence of these properties? I suspect this feeling of awe is what drives some people to make mathematics their life.
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This is the type of thing that fascinates me about mathematics: things that reveal fundamental properties of number, regardless of the base used. I find such things fascinating and mysterious, moreso than just about anything. It leads me to ask what is number? How does it exist? Why and how do mathematical properties such as this arise? And what is the mode of existence of these properties? I suspect this feeling of awe is what drives some people to make mathematics their life.
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Quoss
My code generates a slight bias that seems to grow as you test more consecutive, mine is just a single digit issue.
two3[n11_]: =
Catch[Module[-inp = n11, mem1 = 0, mem2 = 0, mem3 = 0, dat1 = ---,
mem1 = Quotient[inp, 2];
mem2 = Mod [inp, 2];
If[mem2 == 0, mem1 = mem1 - 1];
mem2 = Mod [mem1, 3];
mem2 = If[mem2 == 0, 3, mem2];
Throw[mem2];
]];
tab = Table[two3[Prime[t]], -t, 1000000, 2000000-];
-Count[tab, 3], Count[tab, 2]-
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My code generates a slight bias that seems to grow as you test more consecutive, mine is just a single digit issue.
two3[n11_]: =
Catch[Module[-inp = n11, mem1 = 0, mem2 = 0, mem3 = 0, dat1 = ---,
mem1 = Quotient[inp, 2];
mem2 = Mod [inp, 2];
If[mem2 == 0, mem1 = mem1 - 1];
mem2 = Mod [mem1, 3];
mem2 = If[mem2 == 0, 3, mem2];
Throw[mem2];
]];
tab = Table[two3[Prime[t]], -t, 1000000, 2000000-];
-Count[tab, 3], Count[tab, 2]-
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Rosie
2: 35 -If that's the case, then we're looking at prime gaps less than 10-. The explanation you tried to refute can't be dismissed that way. Where p=10j+9 is prime, -it doesn't matter which decade- contains the next prime >p. Let's say it's the decade of 10k. Then we're given that every integer between p and 10k is composite. The next prime is -still- most likely to be 10k+1 and least likely to be 10k+9. Even if it's unlikely that 10k is just 10j+10.
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2: 35 -If that's the case, then we're looking at prime gaps less than 10-. The explanation you tried to refute can't be dismissed that way. Where p=10j+9 is prime, -it doesn't matter which decade- contains the next prime >p. Let's say it's the decade of 10k. Then we're given that every integer between p and 10k is composite. The next prime is -still- most likely to be 10k+1 and least likely to be 10k+9. Even if it's unlikely that 10k is just 10j+10.
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GlowstoneLove
3, 7, 9, and 1.
The number of rings in Lord of the Rings.
''Three'' Rings for the Elven-kings under the sky,
''Seven'' for the Dwarf-lords in their halls of stone,
''Nine'' for Mortal Men doomed to die,
''One'' for the Dark Lord on his dark throne
In the Land of Mordor where the Shadows lie.
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them
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3, 7, 9, and 1.
The number of rings in Lord of the Rings.
''Three'' Rings for the Elven-kings under the sky,
''Seven'' for the Dwarf-lords in their halls of stone,
''Nine'' for Mortal Men doomed to die,
''One'' for the Dark Lord on his dark throne
In the Land of Mordor where the Shadows lie.
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them
reply
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