William M. Briggs

Statistician to the Stars!

Paper Finds Pattern In Primes: What About Randomness & Chance?

From the paper.

From the paper.

Here’s the headline in Nature: “Peculiar pattern found in ‘random’ prime numbers“, an article which opens “Two mathematicians have found a strange pattern in prime numbers — showing that the numbers are not distributed as randomly as theorists often assume”.

The author, Evelyn Lamb, probably means by “distributed as randomly” distributed uniformly. Now that phrase “distributed uniformly” is tricky. In the interpretation I think she has in mind, it means laying out all primes—or perhaps all those after 5; the primes 1, 2, 3, and 5 are special cases—the ending digits should be equally represented among 1, 3, 7, and 9. (Obviously, numbers ending in 2, 4, 5, 6, and 8 all have divisors of 2 or 5 and aren’t prime.)

This infinite-uniformity might be true, but that is no guarantee that any finite collection of primes shares the same property. And, in fact, most finite collections don’t. Here’s one: 117 (it’s still a collection). Not too uniform, that. But since the number of primes is infinite, we can collect infinite subsequences of primes and check their properties.

Before that, what about distributed? What a tricky word! It’s appropriate in a way. Some thing, or possibly things, caused the universe to be in such a way that prime numbers have the properties they do. What was this thing (or things)? On that subject, mathematicians (and scientists) are silent. In any case, some thing caused the primes to be distributed in the fashion they are. Our knowledge of the distribution is not the knowledge of the cause of the distribution.

Epistemology is not ontology, yet the two are often mixed up. Lamb goes on to say, “number theorists find it useful to treat the primes as a ‘pseudorandom’ sequence, as if it were created by a random-number generator.” Random means unknown and is thus always a misnomer when used in phrases like “random-number generator”. These are deterministic algorithms with known inputs and fixed, known outputs. Yet they’re called “random”. Why? Because, their writers turn a blind eye to the algorithm used and ask how people who don’t know the algorithm see the sequence: can these strangers find any way to predict the sequence, even imperfectly? If not, then the sequence is called (unfortunately) “random”. If they can, then it isn’t.

The paper in question is “Unexpected biases in the distribution of consecutive primes” by Robert J. Lemke Oliver and Kannan Soundararajan. It opens by remarking that Chebyshev had long ago noted that in the first million primes, a certain property “seem[s] to be slightly preferred”. The property was found in only 499,829 of the first million and not 500,000 as uniformity would suggest. Or as “randomness” would suggest if by that word we meant non-predictabile.

The authors modified the property and found that this modification showed departures from uniformity for a finite sequence (the first hundred million). Of course, departures in finite subsequences, as we saw above, may or may not be interesting. What makes the authors’ work important is that they prove, given modest assumptions, that the departures from uniformity hold in infinite sequences (for some but not all the properties they checked). (The modest assumptions are not known to be true, but everybody believes them. Also interesting is that this paper is not peer-reviewed, yet its argument still is treated worthy of discussion: amen to that.)

The authors say:

Despite the lack of understanding of [the certainty property], any model based on the randomness of the primes would suggest strongly that every permissible pattern of r consecutive primes appears roughly equally often;

This equates “randomness” with uniformity. Understand that unpredictability has degrees. If all you know is that some proposition is “normally distributed” (with fixed parameters), then you cannot predict the precise value of that proposition, but you do know that some intervals have greater probabilities than others. This is a departure from uniformity and is, in a weak sense, more predictable.

That same kind of weak predictability is what has been found for some properties of sequences of primes. Lamb summarizes “it would seem that this is because gaps between primes of multiples of 10 (20, 30, 100 and so on) are disfavoured.”

That “disfavoured” is an interesting choice of words. It acknowledges that whatever it was that caused primes to have these newly discovered (or any) properties liked, for whatever reason, the departure from uniformity. These are words describing the actions of intellect—and they are not out of place. For some intelligence had to create. It cannot have been “randomness”, which is not a physical thing, which created “randomness.”

32 Comments

  1. Two mathematicians have found a strange pattern in prime numbers — showing that the numbers are not distributed as randomly as theorists often assume

    Well, yeah, patterns tend to make things more predictable which is the antithesis of random.

    That “disfavoured” is an interesting choice of words

    Or it could be a manifestation of that cutesy tendency to say things like “the ball wants to roll down the hill” when, in fact, the ball is compelled to do so regardless of what it may want. Most people wouldn’t think of the ball as having made a choice. There was a web site once (maybe still is) railing against these usages.

  2. Briggs

    March 16, 2016 at 9:31 am

    DAV,

    Language about the ball is clearly metaphorical, but as I hoped I pointed out, language about what caused this pattern of primes must be literal. I don’t think Lamb understood that; her choice of words was fortuitous. (Get it? Get it?)

  3. Briggs,

    I see that physical events have a cause but how do mathematical relations have one? Isn’t the pattern of primes only a consequence of the definition?

  4. bah

    Multiplication is a human derived shortcut. Nature adds, it doesn’t multiply – and the primes are therefore what they obviously are: the numbers that can’t be reached by integer multiplication. This seems stupid to me, and I think it probably is – but if what is true in math is simply that which can be shown to be reducible to counting on your fingers, then this may be a case of the stupidly obvious being obviously correct.

    i.e. given only the positive primes 0 to a sub i, you can’t make ( a sub i+1) by multiplication and therefore the breaks (ai – (ai-k)) determine where the next one is – completely predictable with nothing random about it.

  5. Briggs

    March 16, 2016 at 10:12 am

    Scotian,

    In part, yes. The definition allows us say what a prime is. But the resulting pattern still has to be there for some reason. That is, it, or any pattern in numbers, has to have been caused. By what? There has to be a reason the universe is this way rather than another. It can’t have been nothing or “randomness” or “chance” that was the cause. So, we look elsewhere.

  6. I can’t do the math in the original paper. I don’t understand infinity, or ‘an infinite series’. Paul Murphy reminds me that I didn’t understand a prime as ‘an integer that can’t be arrived at through multiplication’.

    I don’t know if I even want to know everything else I don’t know.

  7. Why can’t the definition be the “cause”?

  8. I would argue the definition is the cause. If it weren’t for the definition, it wouldn’t be prime. However, if they follow a pattern then there must be other definitions which are also causes in the sense that causes allow us to predict. However, I would think these causes tell us more about the pattern. Even numbers follow a pattern but does their evenness cause the pattern or does the pattern cause their evenness?

  9. I was by two researchers (doctors? I am not sure, they looked awfully young and were afflicted with upseak) and one said to the other, presumably about a clinical trial, asking about the implications of “two or three [getting] better randomly.” She struggled with finding the words to describe the cases that cannot be attributed to the drug therapy. She was aware of the factors that could affect the results, but she was not equipped with the vocabulary she needed to describe what she thought was going on.

  10. pattern exhibits evenness and evenness exhibits pattern
    Cause is separate.

  11. Cause must come first
    pattern is an effect of a cause.

  12. Yes, a pattern is the effect of cause. But a definition seems to be what causes some patterns, as in the prime numbers.

    Look at measurement—metric versus SAE. I would think we could find patterns in measurements done both ways, but the patterns would not be the same in each measurement type. The object measured does not change size, only the definition of the measurement unit changes.

  13. Cause must come first. pattern is an effect of a cause.

    Two things can be a mutual cause. It’s more proper to say effect must follow (vs. lead) a cause. However, cause merely is something that can be used to predict. You can use a pattern to predict. X causes Y. Why? X –> pattern –> Y. Stuff between X and Y is the reason. Regardless of how you think about pattern, it is effectively in the causal chain.

  14. Briggs,

    “But the resulting pattern still has to be there for some reason. That is, it, or any pattern in numbers, has to have been caused.”

    Reason, cause? I think that there is a confusion of definitions here. In the physical realm the hammer blow causes the nail to penetrate the wood. This cause and effect can be repeated for a series of nails. In mathematics there is no cause and effect, there is only the structure of mathematics. Proofs of theorems based on premises is not cause and effect. If you could find a formula to predict the prime number pattern you might call that a cause, but I wouldn’t.

    “There has to be a reason the universe is this way rather than another.”

    Mathematics is not the universe. There may have been different ways the universe could have turned out but the same mathematics would be found in all. Isn’t this what you are saying in the Sunday Series? The properties of God do not depend on the practicular universe we inhabit. This brings up a philosophical conundrum: can God create a universe without prime numbers?

    Your complaint of the use of the term “random” is a different issue.

  15. “In any case, some thing caused the primes to be distributed in the fashion they are.”

    No. Primes exist only in a human mind; they have no natural existence. They exist in a human mind only because of the constraints created to permit primes to exist.

    We create a RULE, and from that rule comes primes. Thus, a human created primes out of the rule that a human created.

    The rule is (1) all natural numbers (positive integers) that can only be divided by either “1” or each by itself.

    Suppose you see a tree branch, and on it are three apples. Does the tree know or care that three is special, four is not, five is special but not eight or nine? Do the apples care? Did any force in heaven or on earth decree that 3, 5, 7 but not 9 is special? That is unlikely.

    Another rule produces the Fibonacci series. Interestingly, this can be found in nature.
    http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm

    But even there it isn’t exactly a rule, it is an observation that leads to the rule, rather than a rule that leads to the observation.

  16. re: Prime Numbers

    Have you read the book: Calculate Primes-Direct Propagation Of The Prime Numbers (Paperback) by James McCanney (Author), ISBN-13: 978-0972218665, 2007?
    Available from:
    McCanney is the guy who in the late 1970s predicted that comets would be dry rocks instead of “dirty snowballs.”
    Dan Kurt

  17. ‘Two things can be a mutual cause.’
    Yes, cause can be complex.

    Evenness and pattern are descriptive of effect which followed a cause even if the cause is unknown. Until a thing is fully understood it can’t be predicted.

    Michael 2
    Of course numbers are what allow us to discover the universe. They are necessary to do most science, music and even art. Of course if God created everything he created the numbers as well.

    How about the puffer fish that makes a pattern or the pattern in a bee hive or flower petals which are usually odd.
    There is every reason to expect patterns. Patterns imply cause and causes a process. Process implies design.

  18. “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”
    “All the Integers, the Dear Lord has made; all the rest is Man’s crafting.”
    –Leopold Kronecker

    One of the questions uncovered by the interesting discussion in these Comments is whether ‘Prime’ is a man-made Rule, and therefore that what we ‘see’ in a prime number series is [merely(?) solely(?)] the computational implication-through-time of our own ‘Man-crafted’ Rule: Paging Stephen Wolfram; Stephen Wolfram, please pick up the white courtesy phone.

    And of course, Wolfram goes much, much further than this; I guess to a ‘computational pantheism’, if I can coin a phrase.

    Whatever the implications to Cause, Messrs. Oliver and Soundararajan (and much thanks to Matt for describing their research) appear to have discovered something real, and unanticipated, about the sequence of Primes. And that is just cool in itself.

    And sometimes, maybe, that’s enough.

    A famous rabbi lay on his deathbed, his students around him; the brightest, nearest him; the dullest, far away at the foot of the bed. Seemingly near his last breath, the Rabbi whispered something that only the very smartest pupil, leaning over close, could hear.

    “What did he say? What did he say?” all the rest whispered. Whispered the brightest to the next-brightest, “He said, ‘Life… is like a cup of tea.'”

    This got whispered down the line, reaching at last the dullest student, who said, “I don’t understand this. Please ask the Rabbi, how is Life like a cup of tea?”

    Each student, both aghast, and suddenly realizing that he didn’t understand it either, whispered the dullest student’s question up the line, until at last it reached the ears of the Rabbi’s very brightest prodigy.

    Who didn’t know the answer, either.

    So this student, leaning close and respectfully, whispered to his dying master, “Rabbi: How is Life like a cup of tea?”

    At this, the Rabbi’s eyes suddenly opened. There was a silence, a silence in which it seemed the very destiny of the world hung in the balance.

    “So — it’s not like a cup of tea,” the Rabbi said.

  19. Until a thing is fully understood it can’t be predicted.

    Not sure about that. Depends on what you mean by “fully understood”. One can predict the next sunrise without knowing why it happens. Same with starting my car with a key. Saying X causes Y is itself a pattern.

    All that’s needed is a pattern and perhaps where in the pattern you are if necessary. Does knowing this mean you “fully understand” what you are predicting? Certainly, with better understanding, whatever that means, one can refine the pattern for better precision or accuracy.

  20. Yes Dav, you are right full understanding is unnecessary for predicting a pattern. I have no idea why I even thought it! I’ll think about it tomorrow.

  21. Joy wrote: “Evenness and pattern are descriptive of effect which followed a cause even if the cause is unknown.”

    These effects exist, or are “instantiated”, in your mind. Evenness is merely the quality of being divisible by 2. What is the name for being divisible by 3? by 4? by pi? There’s probably names for being divisible by every and any integer, fraction or irrational number; and if not, you can create it.

    “Of course numbers are what allow us to discover the universe.”

    Who is “us”? I discover the universe by extending my senses, primarily optical, through binoculars and telescope. Numbers don’t exist. The concept exists; but until it is instantiated as a number of things they serve no purpose other than as a mental exercise. Can you place three in my hand? No. You can place three marbles, but not “three”.

    “They are necessary to do most science, music and even art.”

    Numbers are used for communication. A designer wishes to build a house 14 feet high for instance, and communicates it. But it need not be feet. It could be cubits in which case it will be a different number for the same thing. The possible measuring units are infinite, and therefore the numbers that correspond to a thing also infinite in variety while all meaning exactly the same thing, a physical distance. For any physical thing exists an infinite variety of possible words and measures, and for any measure, an infinite variety of things. These words, “three” for instance, take meaning only as an agreement, a convention, and I teach my children how to interpret “three” so that when I ask for three apples, I will get three apples unless of course my child is being rebellious in which case there’s no telling what I’ll get.

    Geometry allows for many precise operations without numbers, such as dividing a long piece of wood into exactly two pieces of equal length. It can be done geometrically easier and with greater precision than using a tape measure and trying to calculate 1/2 of 95 and 1/2 inches.

    Music does not require numbers; but numbers can help communicate music in a notational form that is not the music itself. The ratio of harmonies is not a number; it can be expressed as a number but the harmonic frequency ratio exists regardless of any number. What is actually pleasing is having, for instance, two waves of one pipe and three waves of another pipe at the same time. But while I describe the phenomenon using numbers for communication, the actual phenomenon does not use or require numbers.

    Within art, numbers can be used to explain certain features that some people find pleasing such as the Golden Ratio.

    “Of course if God created everything he created the numbers as well.”

    I suppose I could blame God for the cookie crumbs on my desk.

    When I go to the beach, and create a sand castle, I am its creator. Until I created it, it was not there and it did not exist. The sand existed, but did not take form until I formed it.

    Obama would assert that I did not build that. Perhaps you will too.

    “How about the puffer fish that makes a pattern or the pattern in a bee hive or flower petals which are usually odd.”

    I do not understand “how about” questions. Bees do not know they are making a pattern. They are very likely trying to make round cells; but it is the nature of geometry that when you press round cells together they form hexagons.

    “There is every reason to expect patterns.”

    Expectation is the only reason. If you don’t expect a pattern you won’t see one. People with very strong expector mechanisms see patterns in clouds. I do not. Is the pattern in the cloud? No, it is in your mind. If it was a property of clouds then everyone would see it. I see white when looking at cumulo-nimbus clouds on a sunny day. That’s all. No fluffy sheep, no horses or turtles.

    The human mind has a library of expectations, more for some people than others. But where you don’t have a “recognizer” you won’t even see a thing. Brain-damaged people are quite interesting. I watched a program some years ago on TV; a person could identify and name a hammer on being shown the hammer, but turn it 90 degrees and now he had no idea what it was. It is a very common part of intelligence testing to show you an object and then you select from four similar objects; one is that object rotated in space and the other three aren’t that object. The ability to recognize an object despite its orientation is very important to the military (and model aircraft hobbyists).

    “Patterns imply cause and causes a process. Process implies design.”

    Honeycombs are a materials-efficient way to create cells for storing honey and placing bee larvae. It implies nothing by itself; honeycombs arise naturally from soap bubbles or pressing flexible cylinders together. Bees do not and did not need to design a honeycomb as an act of deliberation and committee.

  22. Milton Hathaway

    March 16, 2016 at 9:47 pm

    I only skimmed the comments, so apologies in advance, but is this really a serious discussion of whether God created prime numbers? (Maybe they just evolved . . .)

    Hopefully no one brings up imaginary numbers, lest heads start exploding.

  23. People often confuse rules with constraints, it seems to me.

  24. These effects exist, or are “instantiated”, in your mind. Evenness is merely the quality of being divisible by 2. What is the name for being divisible by 3? by 4? by pi? There’s probably names for being divisible by every and any integer, fraction or irrational number; and if not, you can create it.” yes, they are patterns which by virtue of the fact that they are named and can be universally recognised, they exist. They are self evident.
    You can call the thing an abstract concept because you can’t touch it. You can’t touch language either.

    Who is “us”? are you unfamiliar with this term of phrase?

    “Numbers don’t exist.”
    same point really.
    I think you argue that there is only material and nothing else.

    About measurement using string, footprints or inches, these are prescribed units that can be used as templates.
    there’s no ambiguity about measurement.

    “Numbers are used for communication.”
    Yes numbers are language and language self evidently exists. They are used to reduce ambiguity! the irony. I mean only that some love numbers and some don’t. nothing more.
    “Music does not require numbers; but numbers can help communicate music”
    Yes I am one of them I learn by ear because braille music is preposterously complex and you need your hands if you’re playing an instrument.
    However, take most musicians music away and they’re completely lost. So they use the language as a tool. the tool is visible on the paper even to me. Numbers or patterns at least which are universally experienced aid memory of the writer and the reader. Self evident tools.

    “I suppose I could blame God for the cookie crumbs on my desk.”
    Now you seem to be referring to free will to create a thing.
    The argument I make is that the patterns were there to be discovered. I know YOU caused the cookie crumbs because you like cookies.

    “When I go to the beach, and create a sand castle, I am its creator. Until I created it, it was not there and it did not exist. The sand existed, but did not take form until I formed it.”
    The sand castle is a thing though, we are talking about numbers or language.

    Obama? what?

    “Bees do not know they are making a pattern. They are very likely trying to make round cells; but it is the nature of geometry that when you press round cells together they form hexagons.’
    Well we don’t know what bees know but I’ll agree for now that they don’t know a single thing. The bee moves in a certain way and creates a hexagon. The cells are hexagonal not round tubes squashed. It has to do with the way they move and if your assumption was correct about the squashed circles the frames would not be evenly squashed, especially when the honey would weigh more one side than the other and when the comb is incomplete the shape is the same uniformly hexagonal.

    Expectation is the only reason. If you don’t expect a pattern you won’t see one. Well not the only reason but I understand what you’re saying.
    It would be silly to look for something you don’t expect to see wouldn’t it?

    The brain is a fascinating thing and when it goes wrong it reveals something about how memory and language are mapped. In cases of dysphasia (speech or language defects)
    Patterns are noted with certain patients unable to recall names but are otherwise unaffected. Others speak in what sounds like a normal way but with words completely muddled like you took a bag of words tipped them out and made rhythmic dynamic sentences out of them. Sometimes you can understand what they mean by their tone and rhythm. These are patterns of speech defect and relate to where the, usually CVA, has happened.
    In case of lack of recognition this is referred to as visual neglect when people don’t recognise one side of their body and yes it can be even more localised.

    Bees designing or ‘knowing’, I doubt it too. It still implies a design. This design must be part of their original programme. For day to day musings I like to think they’re clever, I do suspect strongly this is not the case.

  25. Back in old days, when the Internet wasn’t really a World Wide Web, we filed a new paper as technical report or working paper. It would be sent out colleagues of the same expertise at other universities for comments, in the hope that our colleagues would catch mistakes, make suggestions and possibly collaborate to work out new ideas. And sometimes, it is an evidence of “who did it first!” Time has changed. Now we share via ArXiv.org or email.

  26. I think the “randomness” of prime numbers means the location of prime numbers within the integers seems random or lack of patterns.

    The authors examine the number/frequency/distribution of primes that meet certain specifications or patterns of interests. The frequency can be derived for a finite case via computer. It is not random. For example, e.g. the number of primes that has a remainder of 1 when dividing by 3 among the first million primes is π(x0; 3, 1) = 499,829. Counted. Observed. Exact. Not random.

    No need for prediction in finite case.

    The authors derive asymptotic laws (approximations) of the frequencies for considering more elaborate patterns (not a modification at all. Note that π(x0; 3,(1, 1)) +π(x0; 3,(1, 2)) ~=π(x0; 3, 1) in the paper by modifying existing conjectures and show that the bias in the approximation decreases as the number of primes number increases.

    Approximations!

    Who or what is the cause? Perhaps https://en.wikipedia.org/wiki/Philosophy_of_mathematics contains an answer.

  27. When I said,
    Until a thing is fully understood it can’t be predicted.

    I was thinking about prediction of complex things which can only be predicted if all the initial conditions and mechanisms are known.
    Patterns can obviously be predicted but this is not to know everything about what is causing the pattern.

  28. Prof. Briggs – I don’t think it makes sense to speak of the primes or their pattern having a cause unless you can say something about what their non-existence might look like. Without that, you can say that God caused them, but this is utterly uninformative.

  29. Wwhat about the puffer fish?
    This little puffer fish takes nine days to build this pattern and decorates it with sea shells.
    If it isn’t what the art critic, the female puffer fish, had in mind he starts again.
    She expects to see a better pattern, evidently.

    https://www.youtube.com/watch?v=yaPmYYWsixU

  30. I had challenged this uniformity a while back. The full comment and somewhat lengthy argument is presented here:

    https://terrytao.wordpress.com/2007/04/05/simons-lecture-i-structure-and-randomness-in-fourier-analysis-and-number-theory/#comment-428732

    The use of 4 sequences instead of an aggregated number line also sheds some light onto the twin prime conjecture.

  31. “Life is not like a cup of Tea!”

    My mother visited this weekend and described for me her most recent success, which was the completion of a puzzle entitled “women in gold”. While doing this puzzle, she took time out to go visit friends who were much better and doing puzzles. She watched them and learned some methods..

    I asked her “What did you learn?”

    She looked at me and said, “I don’t know how to say it.”

    EXACTLY!

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