More on “I could care less about the twin primes conjecture”

As part of a discussion about research retractions, I remarked that I could care less about the twin primes conjecture.

This got some reactions in comments! Dmitri wrote:

I think it’s refreshing that Andrew doesn’t care about the twin primes conjecture. After thinking about it for a few seconds, I realized that I also don’t care about the twin primes conjecture.

It’s kind of interesting to think about what sorts of unanswered questions you actually care about. “Is there life on other planets?” Definitely. “What does Quantum Mechanics mean?” Totally. Twin primes, meh …

From the other direction, Ethan Bolker and Larry Gonick were disappointed, with Ethan writing, “Andrew did mathematics before he did what he does now and I thought some of that curiosity would remain.” Adede followed up with, “I find it interesting that someone can care about whether sqrt(2) is a normal number but not care about the twin primes conjecture. It can’t be a pure vs applied thing, both of them seem equally devoid of real-world applications (unless I am missing something).”

OK, so where are we?

First, I’m a big fan of the Cartoon Guide to Statistics, so if Larry Gonick is disappointed in me, that makes me sad and it motivates me to try to explain myself. Second, hey Ethan, I still have curiosity about mathematics, just not about the twin prime conjecture! For example, as Adede notes, I’m curious about the distribution of 0’s and 1’s in the binary expansion of the square root of 2, and that’s pure math with no relevant applications that I know of.

So here’s the question: Why do I care about the distribution of the digits of sqrt(2) but not twin primes?

I’m not really sure, but here are some guesses:

1. The distribution of the digits of sqrt(2) has a probability and statistics flavor; it’s a search for randomness. I’m interested in randomness.

2. Back when I was in high school and did math team and math olympiad training, there were two subjects that were waaay overestimated, to my taste: number theory and classical non-analytic geometry. We got so much propaganda for these subjects that I grew to hate them. A certain amount of number theory is necessary—factorization, things like that—and, yeah, I get that there are deep connections to group theory and other important topics, as well as connections to analysis. I’m glad that somewhere there are people working on the Riemann hypothesis, etc. But the twin primes conjecture, the 3n+1 problem, etc.: I get that they’re challenging, but they’ve never really engaged me.

Explanation #1 can’t be the whole story, because I also find questions about tilings to be interesting, even when no randomness is involved. And explanation #2 isn’t the whole story either. So I don’t really know. Maybe the best answer is that my understanding of mathematics is sufficient for me to understand lots of things in statistics but is not deep enough for me to have any real sense of what makes these particular problems difficult, and so my finding one or another of these problems “intriguing” or “boring” is just an idiosyncratic product of my personal history with no larger meaning.

To put it another way, when I tell you that the Fieller-Creasy problem is fundamentally uninteresting or that the so-called Fisher exact test is a bad idea or that Bayes factors typically don’t do what people want them to do, I’m saying these things for good reasons. You might disagree with me, and maybe I’m wrong and you’re right, but I have serious, explainable reasons for these views of mine. They’re not just matters of taste.

But when I say I care about the distribution of the digits of the square root of 2 but not about the twin primes conjecture, that’s just some uninformed attitude for which I’m not claiming any reasonable basis.

21 thoughts on “More on “I could care less about the twin primes conjecture”

  1. I like Jazz, blues, bluegrass and the Grateful Dead, but mostly dislike hip hop or Taylor swift or grunge . There’s nothing inherent to that except aesthetics. It’s kinda similar I suspect. Number theory never got me very excited but symbolic logic and programming language design is right up my alley…

  2. Why should the twin prime conjecture even be decidable in the first place? As long as no proof is found, this is just one of uncountable many undecided theorems in number theory and even then it’s one with very unobvious applications. Sure, if a proof is found, then the proof itself is probably pretty interesting, way more interesting than the conjecture itself. Until then, there are, however, many easier but interesting problems, so why waste it on twin primes?

    • Proving twin prime undecidable would be way more interesting than either proving or disproving it. There aren’t a lot of simple natural undecidable examples, and the ones which there are, like Collatz-style conjectures, turn out to have reasonable interpretations as programs (Conway invented FRACTRAN to make that clear) so you immediately see why they were undecidable; I definitely can’t see any way in which there being twin primes corresponds to any programming language or programs, so if it was, that’d be a remarkable language.

  3. Number Theory and probability have a close connection. As an example:

    Let ω(n) be the number of distinct prime factors of x. A theorem of Landau says that for N large, then for randomly selected positive integers less than N, ω-1 has a Poisson(log log N) distribution. This statement holds in the limit as N goes to infinity.

  4. The thing I care about is people saying

    “I **could** care less about X” when what they mean is “I **could not** care less about X”. The sentiment is the exact opposite of the words chosen. Your caring is at it’s minimum. You could not care less about twin primes.

    I know. I know. Language evolves. But, what if math also evolved so that we no longer considered prime numbers to be prime numbers?

    Maybe if I can convince Larry Gonick to tell you this, it might sink in ;)

    OK – I have other windmills to tilt at.

    • Well “i could care less” just means that caring is not at a minimum. “I could not care less” means caring is at a minimum. I’m not sure which Andrew really means, although I suspect it is not the latter (surely there is something he cares less about than the twin primes conjecture. For me, it is pretty low on my caring list, but certainly not the minimum. At the same time, saying “I could care less” really doesn’t say much since it could be a statement that caring is at the maximum, which as suggested here, is the opposite of the intention. I think we need a validated quantitative scale of measurement, such as “how much is Andrew willing to pay to read about the twin primes conjecture?”

    • For Andrew, more interesting things arise daily. How much he cares about Twin Primes is still declining, and probably will be for the foreseeable future.

  5. I tried to contribute to this blog of today, but instead it wound up in its predecessor of December, 2022. So, here is my second attempt to set things straight regarding the vital issue of this decade:
    &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
    &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

    Andrew states, “I could care less about the twin primes conjecture.” BUT, should he have used “I could NOT care less about the twin primes conjecture”?? I found this web site
    ——————————————————–
    https://www.grammarbook.com/blog/definitions/could-or-couldnt-care-less/

    “When someone uses either version of this expression, they are applying sarcasm to convey they have no concern or preference. Therefore, the technically correct version is couldn’t care less.

    It is easy to see why this has to be so when you further examine the thought. Suppose Jim has been asked what movie he would like to see. If he responds “I could not care any less than I already do,” he would be lengthy but clear. On the other hand, if he were to reply with “I have fewer cares,” the meaning would be lost.

    In this way, saying that you “couldn’t care less” is like saying you care “less than nothing.” It’s an obvious exaggeration, but the meaning and emphasis are straightforward.”
    ——————————————————
    The website concludes with the suggestion that either phrase is to be avoided in serious discussions and relegated to “the occasional blog post”:

    ################################################
    “Why We Could Care Less About This Rule

    Hopefully the explanation above clarifies why “couldn’t care less” is better than “I could care less” in conveying the same thought. If you keep using the non-preferred phrase, though, you will probably be in good company.

    The two versions of the given statement have been used so often that both are well understood. In addition, because they are based on informal slang and sarcasm, neither expression will be found in serious reporting or academic work. That means you are more apt to find these constructions in everyday speech and the occasional blog post.”
    #################################

    For more hairsplitting on the subject,

    https://www.merriam-webster.com/words-at-play/could-couldnt-care-less:

    “Both could and couldn’t care less are informal, and so you are unlikely to use either one in formal writing. If you have need of using it in some other context, and would like to avoid alienating some portion of your audience you should stick with couldn’t care less. And if you can’t get past some people continuing to use could care less, and the fact that there’s nothing you can do about it, you may console yourself with the notion that at least they are not saying ‘I could care fewer.’ “

  6. > o here’s the question: Why do I care about the distribution of the digits of sqrt(2) but not twin primes?

    > I’m not really sure, but here are some guesses:

    > 1. The distribution of the digits of sqrt(2) has a probability and statistics flavor; it’s a search for randomness. I’m interested in randomness.

    Well the distribution of primes is also pretty random! that’s why a lot of prime number theorem is statistical in nature

  7. That’s sweet that everyone is so understanding and accepting of others’ views, but I buy it.

    Part of this comes from my personal history including a PhD in math, where I got to witness way too many math-is-so-beautiful statements from my fellow students. There are some mathematical areas that I admit have pleasing attributes which may as well be described as beautiful, but I don’t think I encountered a single topic in 5 years of grad school that didn’t have *somebody* crooning about how beautiful it was. I mean, surely we can agree that some topics are just grubby, boring tools, right?

    While the math-is-so-beautiful trope is sometimes deployed in a meaningful and earnest way, it is also very often deployed in much lamer ways. The worst is a sort of insecure flexing, where someone is trying to say “this isn’t so hard for me, I actually finding is pleasant”. The second is when someone is near the end of their intellectual capacity and they’re having a sort of mind-blown experience with the parts they can’t quite hold in their heads. Or at least, that’s how I usually interpreted the statement.

    Finally, the reason the twin-primes conjecture is objectively boring is that the it at least appears to be saying nothing more than infinity is so big that more or less any specific finite pattern is bound to repeat infinitely many times within it. Mind. Not. Blown. Yawn…

  8. Andrew

    Thanks for the response. I think you’ve explained your “meh” well enough, though (as @Elias comments, there is lots of probability in the distribution of the primes.

    Even though you could or couldn’t care less about the twin prime problem, you might enjoy
    Apostolos Doxiadis’
    Uncle Petros and Goldbach’s Conjecture, about another number theory conjecture even less central than the twin prime conjecture.

    And, in the shameless self publicity department

    Bolker, Bolker and Bolker:

    29 and “29” in base 29 = 67 are both prime. We explore that curiosity and find connections to deep questions on the distribution of the primes: the prime number theorem, Dickson’s conjecture, and Zhang’s bounded prime gap theorem.

    https://www.tandfonline.com/doi/abs/10.1080/0025570X.2020.1704613?journalCode=umma20

    PS Might we please have an editor with a gui for html markup? I think there are some for wordpress sites and I think yours is wordpress powered.

  9. Hey Andrew, not necessarily trying to change your mind, but you mentioned that you like problems with an underlying prob/stat flavor, and the twin prime conjecture (as well as many other open problems in number theory) can in fact be motivated by probabilistic heuristics!

    The prime number theorem states that the number of primes less than or equal to x is asymptotic to x/ln(x). Harald Cramér (of Cramér-Rao fame) inverted this reasoning and conjectured that the primes should behave in many respects like a random integer sequence with local density 1/ln(x). i.e. if we select a random subset S of the integers with every integer k being included in S with independent probability 1/ln(k), then generic properties of S should be identical to properties of the primes.

    Going off of the so-called Cramér random model,
    P(x and x+2 are both prime) ~ 1/ln(x) * 1/ln(x+2) ~ 1/ln^2(x)
    So that, by linearity of expectation,
    E[# of twin prime pairs infinity, so we would expect an infinite number of twin prime pairs.

    The probabilistic reasoning suggests a concrete asymptotic growth rate, a much stronger conclusion than simply the infinitude of twin primes. This is formalized in a much harder conjecture https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture. The conjecture is based off a more sophisticated random model, and so predicts the same order of growth but with a different constant of proportionality. The precise constant is gotten by assuming that x (mod q) is uniform and independent for different small primes q.

    One last note… the conjecture being due to Hardy and Littlewood is notable as Hardy very much disliked probability. Persi Diaconis has a wonderful essay discussing Hardy and his relationship to probability here https://finmath.stanford.edu/~cgates/PERSI/papers/Hardy.pdf, where he mentions, with reference to this conjecture, that ‘Hardy’s papers give a sophisticated development of conjectured asymptotics which suggest that at least one of the authors was quite familiar with probabilistic heuristics. Here, Hardy was certainly aided by earlier heuristic investigations of Sylvester and others. Hardy refers to what I would call ‘probabilistic reasoning’ as “a priori judgment of common sense” in his expository account (Hardy (1922, page 2)’

    I wish my common sense were as sophisticated as Hardy’s!

    • Seems that two lines of my comment got cut out because of the use of the less than and greater than signs. The omitted part should have been–

      So that, by linearity of expectation,
      E[# of twin prime pairs less than or equal to x] ~ integral_2^x dy/ln^2(y) ~ x/ln^2(x).
      This goes to infinity as x goes to infinity, so we would expect an infinite number of twin prime pairs.

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