Teenager solves stubborn riddle about prime number look-alikes
quantamagazine.org
quantamagazine.org
This kid walked straight up to the board and explained how you can design a computer program to factor any polynomial equation string input to it, and in fact had implemented a polynomial equation factoring program while the teacher explained how to factor simple quadratic equations.
Since then, I don't feel bad if someone achieves more than me, because clearly there are some people out there that are born to solve certain classes of problems (maybe their brain structure is better for those, or something, who knows).
But we also need to make sure that we respect the humans who possess those brains too...
This growing up thing is hard.
Due to having a reading level 10 years ahead of my peers, and already having learned math concepts at least 4 years ahead of them... at age 8... I was put 3 years ahead of my peers in school until the teachers realized the other kids were having none of it.
They put me back in the 'proper' grade, because 'I' somehow wasn't going to develop social skills well enough in such an environment.
They might not have been wrong, but they would have been more correct to tune in the little buggers causing me grief. Or at least let me tune them in...
That said, I should add that 2 of those students ended up running into me again years later. They apologized, which was nice; but then offered me the chance to go do lines of coke with them...
I declined, and went on with my life knowing doubly that my instinct about them was correct the whole time. Wastes of skin.
I would flag your comment if I had the authority. This is not acceptable language for this forum.
Most are still struggling to get over it decades later, myself included.
Quite frankly, I think we all should be banding together to sue our old schools over this stuff.
I could read since I was 4. When others at school were reading one word per minute spelling letter by letter, I finished the whole article and then continued to another and another. Result? I got a teacher's note (a big deal where I live) almost every lesson, and bad grades. And the children hated me, probably mostly because of the teacher mocking me for not paying attention. I would give anything to be boosted to similarly skilled children. I was friends with them anyways, never really liked the kids from my grade (but that could've been avoided if I wasn't forced to "learn" with them).
I've never seen a boosted child say it wasn't worth it or that they had much problems being a year/two younger than their classmates. Actually, smaller schools here usually combine 2-3 grades into one anyways and the children are friends alright, so boosting the taught curriculum is really not a big deal (unfortunately the system didn't allow it when I went to school).
I think that in many case, trying to pretend that people are the same when they are not is causing a lot of pain for everyone involved. Holding a conversation with a normal person requires me to expand a lot of energy to not be myself, because if I am I will quickly be hurt and/or rejected. And just like I often hurt people by being myself, other people hurt me by being themselves. Trying to explain it to them has been pointless in all situation, and most of the time makes the situation worse.
Why would any of us have to endure that, when we could instead be with groups of like-minded people and enjoy life more? I don't feel superior to those people, and spending more time with them leads me to develop more negative sentiments towards them. I think the sane option is to avoid them as much as possible, while making sure that I treat them with respect. I can't change fundamentally who I am, they can't change who they are. I can't understand why you advocate for a situation that creates so much suffering for everyone involved.
The problem is putting a child in an environment where they are surrounded by adults all the time. Like being 17 and having your 'peers' be 30 year old post-docs who are getting married and having children is a complete mind-fuck.
Being surrounded by adults all the time is only a problem if you never spend time around actual peers. Nobody should be forced to isolate themselves like that.
What is the solution? I will not spoil it for you, but it does not involve giving up.
Society wasting and discarding undiscovered talent is one of my deepest fears.
SJG is underrated.
She literally couldn’t get her head around a student going to the library, taking out a book on maths and teaching themselves because they where curious.
The UK has a very strange educational system, curiosity isn’t encouraged, you learn like a good little peg in a round hole.
Left a very bad taste in my mouth but at least she did eventually apologise.
Your way sounds better.
They want to encourage self directed learning and I am glad they did. If nothing else, boredom becomes a good thing more often. The seeking begins and with that can often come some real learning and insight as it may be applied.
The pay is shit, so intelligent people decide to pursue other passions instead of teaching.
Also, the profession seems to attract the kind of people who like to exercise arbitrary authority over other people. Not a majority of teachers, but a substantial minority.
Once I realized these things, a lot of negative experiences I had with teachers made more sense. For what it's worth, I also had some great teachers along the way.
Similarly, the IQ test is flawed too. Neither are representative of intelligence or what is required to be a good teacher
That was the feeling I had after one week in a UK school through an exchange program. Because my original exchange partner quit on me after seeing my picture, I got assigned a different one who was 2 grades higher. I was pretty average in math back in Germany, but in that class, 2 years above mine (11th while I was 9th), their math riddle that was supposed to take them the whole week was solved by me in one lesson.
This was around 20 years ago, though.
I have a step-son who is (nearly) 13 and about to start seriously in on his GCSE's - the situation hasn't improved, the non-higher tier papers are just embarrassing, the foundational tier he could have done well by the time he was 10 - he likes maths and I always wished I'd pushed it harder when I was young but a spectacularly shitty home-life ensured while I did decent in my GCSE's I didn't do what I could have done - I make sure he doesn't have the same problem.
Checked with the teacher at the beginning of the year and she said no problem. After rampant cheating, mostly people just writing notes in the programs section, she instituted a policy of deleting everyone’s calculator before any test. but let me keep mine too, I had asked to not delete it and I’d take a class calculator if she insisted.
So I memorised my quadratic-solver-with-intermediate-steps and tapped it back in while we were waiting for the test to start :)
The calculation was not hard, but it was still faster just to run my program and scribble down the steps it spat out than to do the solving manually each time it came up. IIRC I had one for simultaneous equations too.
As a math teacher, I very much like every part of your comment except for this. There are way too many people who decide never to try mathematics because they weren't born to it. I don't believe (and it seems rather insusceptible to proof) that math is something that you are born with a talent for, or not; instead, it's all about how much passion you have for it. Genius is any field is not born of effortless gift; it is born of tireless effort. The real difference is between the people who are willing to put in that tireless effort, and those who do not wish to do so (which is a reasonable choice for something you're not passionate about!).
And. I. Struggled. So. Much. with math classes.
Could never get homework done because I couldn't focus due to undiagnosed (and thus unmedicated) ADHD. On tests it'd take me the whole hour to do three problems out of twenty because I had to re-derive every equation and lemma from first principles before I convinced myself I was doing it right. And then (probably again due to the undiagnosed ADHD) I'd transpose symbols while working the problem or make other working-memory errors.
The best math class experience I had (later in life) was with a professor at a community college who was a military veteran (mentioned because it influenced his attitude toward teaching). He said, "I'm going to spend hardly any time explaining things; we're just going to work the problems." And that's what we did; we just worked problems.
You'd think I'd be bored with that, but it was the opposite. As I think about it now to try to explain why, I realize that other math classes I'd been in the expectation was lecture was for explaining the concepts, homework was for self-directed working the problems, and tests were for demonstrating that. But if you're bright and have ADHD you're quickly bored with the explanation and completely UNable to self-direct working the problems. So the class time was wasted and homework sessions were hell. So this professor using class time to work the problems was just what I needed.
Went back to community college after some time as a web dev, and had a teacher with a booming voice and a gentle attitude, who explained that we were going to be doing a lot of homework. Like your instructor, he'd spend class time working problems, and then I'd go home and do an hour or two of homework every night while it was still fresh. That kicked off a trajectory that resulted in a PhD and a very fulfilling job as a mathematician.
Once I started to really want to understand and write audio processing code, I suddenly had a framework to not only care about Trigonometry, but to actually apply it and to have a tool that would validate my understanding of it. That short feedback loop got me through the subject incredibly quickly.
I was completely blown away when I started reading some Linear Algebra material because I realized that I had essentially been doing it for years without even strictly realizing it. The subject came to me very quickly because I already had a solid "mechanical" understanding of how to implement systems utilizing a particular subset of it.
I think passion for the deeper maths is something most people just have to discover through application, and there just happen to be a rare few who discover a pure love of it very early in life.
In my view, not everyone needs to be good at everything, but I do get that society sets up maths and science as way harder than it has to be. Part of that is how we teach math, too. I hated math until I went to college and roundabout earned a maths degree.
On the other hand, there definitely are people for whom mathematics is much easier than others. I found mathematics effortless at school and even to an extent at university in a way that other people clearly didn't. I did work at the Olympiad type stuff at school I guess but that felt like fun.
I didn't become a Fields Medallist though :-)
Most of the international-calibre mathematicians I have crossed paths with did find a certain level of maths effortless in the same way I did I think, but they also worked hard, although not necessarily relentlessly. Most certainly had other interests outside mathematics. They all really really loved mathematics.
"3xamoke" is "example", right? If I seemed to claim that good mathematicians found mathematics effortless, I definitely didn't mean to! Rather I meant to claim the opposite: that doing truly deep mathematics (or anything) is deeply effortful even for those who wind up excelling most at it; those people are just the ones who feel such a passion that they can't stop putting in that effort.
> On the other hand, there definitely are people for whom mathematics is much easier than others. I found mathematics effortless at school and even to an extent at university in a way that other people clearly didn't. I did work at the Olympiad type stuff at school I guess but that felt like fun.
I don't claim anything as absurd as that there is no difference in mathematical ability between different people, only that (1) I think the relevance of ability compared to effort is overstated, and (2) even to the extent to which ability is relevant, looking at certain people and declaring them as just good at math (a) undervalues the effort that they put in and (b) discourages people who struggle with it, and who think that means that they cannot do well at it.
My algorithm was slow garbage though, so I'm sure your friend's was much better.
I’m confident this is a famous quote. But once I heard it, it kept resonating.
Then years later I watched Bluey with my kids and mom says, “just run your own race” and it all clicked. I’m so thankful that it clicked because I feel liberated from this self-imposed sense that I need to absolutely maximize my time here, which is an impossible task.
I realized it didn't matter. What mattered is that I progressed toward my goal at whatever pace I could muster, not making it a race. Interestingly, this mindset has made me willing to take on much larger and more complicated challenges in general, one day at a time, without feeling overwhelmed by the mountain to climb ahead.
This is the mindset that got me to do so many projects where a younger me thought, “someone smarter has already done a better job at that…”
I now write software for me. Sometimes I publish it.
(Seems like a good site for verifying quotes, it's crazy how many commonly-believed-to-have-been-said-by-X quotes simply never were...)
If you want some advice on how to avoid having your "joy" stolen by comparison, you could always go with
https://quoteinvestigator.com/2017/10/03/life-success/
Though I suppose "Z" these days would logically include "avoid replying to online forum comments..."!
I've seen it phrased better elsewhere, but otherwise yes.
"However much you possess there's someone else who has more, and you'll be fancying yourself to be short of things you need to exact extent to which you lag behind him."
Lucius Annaeus Seneca, Letters from a Stoic
There are many people who are fortunate in material or circumstantial ways, but you see ways in which they’re enslaved by it or it somehow dictates their life. The classic “owned by what your own” paradigm which is eerily common in western culture.
This can have the effect of making you feel fortunate for having less, rather than envious.
I love Seneca. What I haven’t got a good grip on is time. On the Shortness of Life is a bit of a gut punch for me.
exact same experience... there is literally a genetic factor but people are not willing to admit it
True genius is always happening at the crossroads of his interests. Not linear, not monosyllabic, but divergent-recomposing-thing.
I find this difficult to square with the well-known theorem that there is no closed-form solution to polynomials of degree five or more. (Where a "solution" and a "factoring" are, for polynomials, the same thing.)
If you brag about how "you can design a computer program to factor any polynomial equation string input to it" in a class about quadratic equations and your code can't factor x^2 - 2, that's just not very impressive, regardless of your age.
When the solution is irrational, as is common for polynomials, it might be difficult to guess what it is based on knowing an approximation of it.
When the problem posed is "factor this polynomial", it's unheard of. This is the conceptual difference between "needing a number to work with" and "studying math".
I have never been so wrong haha
I’m sorry if I ruined anyone else’s math classes! If it’s any consolation, I ruined them for me too. I lost interest in the whole subject when I recognized the pattern.
1. Fermat's Little Theorem: if p is prime, then b^p = b (mod p) for all integers b. i.e. b^p - b is always a multiple of p. 8^3-8 = 512-8 = 504 = 168 x 3.
2. Is the inverse true? Does b^n - b = 0 (mod n) mean that n is prime? No. Sometimes n is non-prime (like n=561, divisible by 3). We call these n, Carmichael numbers.
3. Okay, so these numbers exist. How common are they? For primes we know they're common. Bertrand postulated (Chebyshev proved) that for any n>1, there is a prime p between n and 2n. That's cool!
4. Is it true that there is such a bound for these pretend-primes? Well, we have an interesting fact that there are x^(1/3) of them below any x, once we pass a certain point (i.e. there exists an X such that there are x^(1/3) of them below any x > X) so that makes us think it could be true! Worth seeing!
5. But what about this common-ness measure like the B-C result for primes? Well, it turns out that it exists. It ain't as pretty as just between straight integer multiples, but the fact that it exists in some shape at all is cool! That's what this kid proved. Absolutely rollicking fact. https://arxiv.org/abs/2111.06963
It does sometimes feel like it's only there to bulk out the article.
The writer can fix the article, or let every reader fix it in his mind.
>> In fact, Larsen’s argument didn’t just allow him to show that a Carmichael number must always appear between X and 2X.
And yet the Wikipedia page says 2821 and 6601 are the 5th and 6th Carmichael numbers, which means there are not between 3000 and 6000 (X and 2X). So is his result actually that one must always exist between X and 2.5X or some other small multiple? If so, what multiple did he prove?
The bound is easiest seen on the arxiv link above, in the abstract. The HN forum software doesn't do math very well.
https://ar5iv.labs.arxiv.org/html/1910.06709
(great work by FAU Erlangen's Michael Kohlhase and team).
Actually: the form that Larsen proved is stronger than "for sufficiently large X, there is at least one Carmichael number between X and 2X".
So with the more specific proof, he also proved the simpler statement that's easier to express.
Daniel Larsen's result here is that there are e^((log x)/((log log x)^(2+d))) Carmichael numbers between x and (x + x/((log x)^(1/(2+d)))) for x>=X (depends on d)
e^((log x)/((log log x)^(2+d))) is >= 1 for all x >= 1.
(x + x/((log x)^(1/(2+d)))) <= 2x
Stronger by being tighter than the (x,2x) bound and being more specific about the >= 1 number of Carmichael numbers
Very frequently, their children will have a scary intuitive understanding of concepts that took my many years to understand (I'm a slow learner; didn't really understand hash tables until my 30s) and then apply their abilities to be in the higher echelons at science in a very young age. I see a similar thing in the children of world-class athletes.
I think society settled on a response of "yeah, but we shouldn't actively do anything about it as a group." It seems to be best left to individuals to seek out parental partners and enjoy the outcomes on a personal level.
I hope there comes a day when humanity is responsible enough to handle precise gene-tailoring of humans without it devolving into a cliche Sci-Fi dystopia. Until then, I'm glad that humanity has settled on "Eugenics could theoretically work but we know we're not responsible enough to do it right, so let's not". It feels like a very enlightened stance: better than pretending selective breeding would literally not work, and certainly better than doing it badly. Why could we come to such a sensible consensus on this topic but fail to do so on so many others?
All that being said, I still want to see what would happen if Michael Phelps and Katie Ledecky had a child together. You know. For science.
Power itself is a bit of a paradox - having some is essential to a free society, but too much ends up corrupting its holders and results in human suffering. The power you describe falls in the latter category, IMHO.
Part of the reason we're not ready for it is that any notion of "societal IQ" is simply not as mature, well-understood, proven, or even useful as our understanding of climate science. Scientists who studied the topic understood the greenhouse effect and the potential for CO2 emissions to result in global warming over 100 years ago. Meanwhile, nobody can agree on whether the Myers-Briggs personality test means anything at all.
We just don't have a way to "rate" humans in a general sense, nor is it obvious that increasing their "rating" would even be a good thing. Selecting for humans with the highest IQ would be like selecting for crops that grow the tallest. Sure, sometimes that's a good measure of general health, but it's so blunt compared to the intricate nuances of how DNA works and the huge variety of outcomes we'd like from our crops. Besides, it's just a weird stance to want to create a "superior human". Why do we want to do that? So they can solve humanity's problems (no pressure!)? So we can have sex with them? So we can "purify our gene pool" (shudder)? So they can live a happier life and avoid disease? Some of those reasons are much better than others.
Yes, messing with the climate has risks, but global climate is a lot less complicated than the DNA of a single human. And we know our climate is headed for disaster if we do nothing, so the risk of not messing with it is much higher. Similarly, we know people with certain genes are doomed to a die young or doomed to have certain major disabilities. Those will (and should) be our first targets of human gene editing, because the risks of not doing anything are higher.
And of course we already are practicing a light form of selective breeding at IVF clinics during embryo selection. Mostly they look for chromosomal abnormalities, but I'd guess there are clinics out there that offer more advanced gene selection (if not now, then soon).
I would argue that it took thousands of years of history and one of the darkest episodes of humanity in the 20th century to truly realize this.
It's a gedanken experiment, but I would expect that people who measure low on IQ tests but are raised immersed in an technical or artistic environment have as much potential to become masters as people with higher IQ, or very nearly so. Further, I think that by increasing the quality of the education system in a country, you can massively increase the intellectual output. There are probably enormous amounts of untapped potential never matched to an equivalent supply of knowledge and skill.
I would strongly expect the opposite. But I agree it would be beneficial to figure out which of us is right.
They might end up tremendously unhappy, of course. It just goes to show how raw IQ is a silly measure to value, let alone optimize for.
If you're just talking about the Flynn effect, then your conclusion is overreaching.
IQ is still very highly heritable and adjusted for the Flynn effect has very little to do with early exposure to abstract reasoning AFAIK (but would love to see those studies).
The fact that something is allowed doesn't make it not eugenics. Choosing your mate for their phenotype is eugenics.
This sort of makes sense to me.
If you're choosing mates at all, it means you're judging the value of other human beings. You're accepting some and rejecting others based on what you think or feel is best. It's just eugenics practiced by the general population according to highly variable personal criteria.
This is not inherently superior to some explicit eugenics program. It's about the principle of individual autonomy.
In my area the schools are providing computers at a young age and spending time teaching STEAM topics, for example. It's not going to make everybody into Nikola Tesla, but the effort to do something about it does exist.
It wouldn't make much sense to report things that were related to upbringing together with genotype, as those are specifically the two things you want to decompose.
> For more than a year and a half, Larsen couldn’t stop thinking about a certain math problem.
It's rare to be able to focus on something for that long without giving up.
Probably something in the 10-15 age range, or 8-10 and originally in Russian
It's bankrolled 100% by this generous billionaire https://en.wikipedia.org/wiki/Jim_Simons_(mathematician) and I hope he continues to fund it
Here, it means the mental stamina to sit down and focus on a single problem for a long time.
Disclaimer: I'm not German (although some ancestors in the 1800s were.)
Given the extreme connectivity of the present, we are also exposed to brilliant minds with incredible capabilities, making us (me at least) feel even more incapable..
I guess it is a lesson for humility.
Good job Daniel, you show us !
Maybe there’s a path now for these kids.
This may be worth reflecting on [0].
Then I figured some hobbies are better suited for my use case. For example science is better suited for me. I somehow was never bothered by the fact that some geniuses in 19th century already knew enough math/physics that I could digest in my life. Science is infinite so however much knowledge someone gained, it's zero comparing to the total amount. I guess I need that kind of comfort to commit to something. Fossil collection is also a good one because every piece I collected is MINE and unique so I don't need to compare with someone else.
I think this is important to consider. Anyone even cursorily interested in science can know (just via public information, libraries, etc.) so much more about "everything-except-very-specific-thing" than any of the 19th century geniuses who knew and understood so much about "very-specific-thing".
Ofc, the geniuses are ultimately the ones responsible for the advancement of the state of the art, but... there's also a lot to appreciate for the rest of us.
The kid grew up with two mathematicians, so I'm sure they knew how to get him in the right direction. Kids also tend to value whatever their parents are doing.
And not to mention that most kids have all the time in the world to do what they enjoy - and rarely have other concerns.
I'm a musician, and grew up playing my instrument from 4 to 8 hours a day, every day, almost all year round. It was pretty much the only thing I did between age 10 to 18. I had very supportive family and mentors, and got lots of chances others at my age did not.
Naturally, I met a lot of other talented kid. Some child prodigies.
One thing I have noticed afterwards: Being at such a high level as a kid - does not necessarily mean you'll go on to do great things later in life. Some kids just grow out of it, lose interest and motivation, and - well - life often just happens. Even among the most talented, only a small % of them end up doing remarkable stuff later on.