Teenager solves stubborn riddle about prime number look-alikes

https://www.quantamagazine.org/teenager-solves-stubborn-riddle-about-prime-number-look-alikes-20221013/

by theafh • 4 years ago

390 204 4 years ago

tr33house• 4 years ago

very impressive to have such fundamental contributions at such an early age. To even know it's applicability to modern day cryptography is also really impressive. All the best to Daniel Larsen!

4gotunameagain• 4 years ago

Getting older, sometimes it can be so tough to accept the fact that people a fraction of your age achieve things you never will.

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 !

jamiek88• 4 years ago

I wonder how many more Srinivasa Ramanujan‘s died in poverty in a rural village?

Maybe there’s a path now for these kids.

lo_zamoyski• 4 years ago

Why is it tough?

This may be worth reflecting on [0].

[0] https://www.newadvent.org/summa/3036.htm

dylan604• 4 years ago

Yeah, I have no problem admitting that there are people much smarter, more talented, better looking, etc than I am. The day that I assume I'm the top of whatever category is a sad sad day, as I must be the only one left.

svat• 4 years ago

Tom Lehrer once said: "It is a sobering thought that when Mozart was my age, he had been dead for two years."

dekhn5• 4 years ago

More Lehrerisms here: https://en.wikipedia.org/wiki/That_Was_the_Year_That_Was

dQw4w9WgXcQ• 4 years ago

The set of things you will never achieve is infinite compared to the set of things you will, best to stay focused on the latter and enjoy your life. Unless of course you prefer drinking from an infinite well-spring of misery.

3638494737542• 4 years ago

Not to downplay any of Daniel’s accomplishment but sometimes it isn’t a “fair” comparison when others started younger with more resources. His father is a distinguished professor of mathematics and his mother is a professor of mathematics. When you have that sort of resources available at a young age and advanced training you’ll probably accomplish more sooner than someone of similar IQ without those resources who started later

paulpauper3• 4 years ago

There are thousands of mathematicians in the US. I am sure many have kids. How many of those kids do even a fraction of what Daniel did even when having every possible advantage? Today, young people have assess to more resources than ever, yet talent is one of those things that resists this trend of egalitarianism seem elsewhere. More resources means that the super-talented will pull way ahead of the untalented or only moderately talented.

munk-a3• 4 years ago

I celebrate people who achieve memorable milestones like this or build something that revolutionizes the world - but there are a plethora of different ways to live life. For me, as long as you're happy and you're putting more happiness out into the world than you're consuming then you're achieving a pretty damn good life. We're not all responsible for the entire world - as long as you're leaving your little corner of it better than you found it then good on you.

paulpauper3• 4 years ago

It's not enough to make the world better or be happier, a lot of people also want acclaim and recognition. Why do so many people apply to Ivy League schools when 50-100 ranking schools can also provide a good education? Status is necessarily scarce.

paulpauper3• 4 years ago

it's not like everyone can win the IQ lottery. Superior ability is unequally distributed.

strikelaserclaw4• 4 years ago

people should measure themselves by how best they utilized their gifts and circumstances to live a life that they are happy with. No point in comparing yourself to some kid who probably inherited an amazing mathematical brain through no conscious effort, moral superiority etc.. of his own.

TrackerFF• 4 years ago

I see lots of users here hammering on about IQ - but that's just one factor.

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.

renewiltord6• 4 years ago

Summary:

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

vorticalbox• 4 years ago

Thanks for this. I find articles like this super hard to read due to the mixing of the topic and all the "back ground".

It does sometimes feel like it's only there to bulk out the article.

renewiltord6• 4 years ago

One can hope that future AI autosummarizers can be aware of our personal level of knowledge!

jbverschoor2• 4 years ago

Lazy writers and/or who get paid per word.

The writer can fix the article, or let every reader fix it in his mind.

phkahler• 4 years ago

But from the article:

> 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?

bibanez• 4 years ago

I only read the abstract and the result is in the same spirit but doesn't say exactky between X and 2X. It's between some more complicated expressions (using logs like usual in number theory)

renewiltord6• 4 years ago

Ah I didn't mean to mislead that it was x and 2x. Corrected to be slightly clearer.

The bound is easiest seen on the arxiv link above, in the abstract. The HN forum software doesn't do math very well.

zeroonetwothree3• 4 years ago

It’s true for x sufficiently large. So there will be some small counterexamples.

jll29• 4 years ago

By the way, if you don't like reading bulky proprietary PDFs, there is a trick: substitute the x in arxiv.org by the digit 5, and you will see the paper rendered in HTML5, e.g.:

https://ar5iv.labs.arxiv.org/html/1910.06709

(great work by FAU Erlangen's Michael Kohlhase and team).

renewiltord6• 4 years ago

Good tip. I should have just linked the arxiv.org page for the article and not the PDF directly https://arxiv.org/abs/2111.06963

vitus• 4 years ago

It ain't as pretty as just between straight integer multiples, but the fact that it exists in some shape at all is cool!

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.

renewiltord6• 4 years ago

Haha very good!

gnubison2• 4 years ago

What is the stronger version?

sagaro• 4 years ago

thanks for the really good and concise summary

ok_dad5• 4 years ago

I knew a guy in high school that carried around a sub-compact notebook and one day in science class we were learning about how to factor quadratic equations (a review of old math we should know) and this guy was not paying attention at all, just typing away. The teacher asked him what he was doing that was so important that he couldn't listen, and to please come up and solve the problem.

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).

Vt71fcAqt7• 4 years ago

And, to add to this, we need to find and nurture these minds so they don't get bored and can grow at their natural speed.

RC_ITR6• 4 years ago

Throughout high school, Kaczynski was ahead of his classmates academically. Placed in a more advanced mathematics class, he soon mastered the material. He skipped the eleventh grade, and by attending summer school he graduated at age 15. Kaczynski was one of his school's five National Merit finalists and was encouraged to apply to Harvard.[17] While still at age 15, he was accepted to Harvard and entered the university on a scholarship in 1958 at age 16.[19] A classmate later said Kaczynski was emotionally unprepared: "They packed him up and sent him to Harvard before he was ready ... He didn't even have a driver's license."[9]

But we also need to make sure that we respect the humans who possess those brains too...

Zamicol• 4 years ago

I don't hear this viewpoint acknowledged enough.

Society wasting and discarding undiscovered talent is one of my deepest fears.

wingmanjd2• 4 years ago

I wasn't as cool as this kid, but I wrote a binomial expansion program in TI BASIC back in high school that I was pretty proud of. Teacher said it was neat, but then banned calculators on our tests/ quizzes after I demoed it.

ok_dad5• 4 years ago

It was a while ago, and my memory sucks for those kind of things, so maybe it was you! Anyways, programming anything in the space of 5 minutes is hard enough for me, so I'm still amazed at that binomial expansion (which I had to look up, myself).

saurik• 4 years ago

I remember my teacher banned the use of software on calculators (not the calculators themselves) but left an exception for me because she knew I was writing all the software I was using and didn't want to discourage me using that to learn about both algorithms and the underlying math.

noir_lord2• 4 years ago

I had a maths teacher who accused me of cheating because we had coursework to solve a particular problem, I solved the problem for that case and then the general case using math she hadn’t taught me even had a pascal program that you could enter parameters and it’d give you the answer.

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.

idunno246• 4 years ago

I wrote programs too. Even had them print out the work.

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.

JadeNB2• 4 years ago

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).

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!).

phaedrus• 4 years ago

What's funny is I was that guy (not literally the same guy, but did similar things) as GP describes who could type in a program implementing the math lesson in the time it took the teacher to describe it.

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.

akira2501• 4 years ago

In school I essentially hated all math beyond Algebra, because it started to become something less tangible and more rote. Which is a sensible way to teach it as a pure subject, but it completely undercuts it's utility and connection to real world problems and the intrigues that go along with them.

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.

ok_dad5• 4 years ago

I mean, I agree anyone can do anything with enough effort, but some people have an easier time. It’s really easy for me to program computers. I routinely am perplexed that people I know can’t understand or see the algorithms and structures I see. That doesn’t mean I’m the best or that others cannot learn it, just that somehow my brain is structured, whether through nature or nurture, in a way that helps me do logic and math required for programming. On the other hand, I cannot learn other languages. I just cannot figure them out. I still worked hard to learn that skill, but I definitely worked way less hard to become experienced at this than so many programmers I know.

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.

jvvw• 4 years ago

Maryam Mirzakhani is a lovely 3xamoke of somebody who didn't find mathematics effortlessly easy as a child.

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.

sudosysgen• 4 years ago

I wrote a similar program in those years about when we were first learning of quadratics. It's not really that I'm very smart, my parents were just math teachers and they'd force me to go through more advanced math books where I learned stuff, and I'd get bored in math class.

My algorithm was slow garbage though, so I'm sure your friend's was much better.

jablongo• 4 years ago

A computer program to factor any polynomial equation string input to it: Do you mean a program to solve for f(x) = 0 using numerical approximation? Factorization has a specific meaning and is not necessarily possible for quintic and higher-degree polynomials / there is no closed form solution like the quadratic formula for n>=5.

THENATHE• 4 years ago

If they’re talking about the quadratic equation, this is likely 8th grade to 10th grade in the US education system. Factoring would be y^2 = 3(ab)^2 + c^2, get y all alone so it’s y = _____

casey2• 4 years ago

Yes, polynomial factorization does have a specific meaning. You should learn it. https://en.wikipedia.org/wiki/Factorization_of_polynomials

ctchocula• 4 years ago

To clarify, there do exist closed form solutions for quintic functions; they just cannot be expressed using rationals and a limited set of operations of additions, subtractions, multiplications, divisions, and root extractions. However, if you do not limit yourself to those operations, there do exist closed form solutions to quintics. See [1] for a few examples.

[1] https://mathworld.wolfram.com/QuinticEquation.html

Waterluvian5• 4 years ago

Someone once told me, “comparison is the thief of joy.”

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.

sho_hn• 4 years ago

Learning a new and at the time somewhat obscure (in my setting) foreign language when I was around 30 taught me this lesson at the time. I did well in the small class I attended with others, but not as well as some of the far younger students with both more free time and perhaps better age-given baseline performance at language acquisition (though this is not entirely clear-cut scientifically).

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.

chatterhead• 4 years ago

I don't know about mathematics, but in orgies this is absolutely true.

wizofaus• 4 years ago

https://quoteinvestigator.com/2021/02/06/thief-of-joy/

(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..."!

khazhoux3• 4 years ago

Someone once told me, “comparison is the thief of joy.”

I've seen it phrased better elsewhere, but otherwise yes.

peddling-brink• 4 years ago

Bluey is a gem.

collin128• 4 years ago

Reminds me of Seneca:

"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

RHSeeger• 4 years ago

I like to use the phrase "I don't need to be better than someone else; just better today than I was yesterday".

WanderPanda2• 4 years ago

Consciously knowing that comparing yourself too much to others is one thing, actually stopping it is a whole different story at least for me. I didn‘t succeed yet. But I noticed that the idea of having 20% personal growth each year is really strong

ducktective• 4 years ago

there are some people out there that are born to solve certain classes of problems

exact same experience... there is literally a genetic factor but people are not willing to admit it

swayvil2• 4 years ago

It's easy to be a genius. All you have to do is give 100% of your attention to your subject, indefinitely.

novosel• 4 years ago

That is the definition of a genius-savant.

True genius is always happening at the crossroads of his interests. Not linear, not monosyllabic, but divergent-recomposing-thing.

strikelaserclaw4• 4 years ago

i mean achievement is at the intersection of luck, talent, and hard work. Knew several people with the raw iq to be great but these people would much rather spend their time doing mental puzzles.

thaumasiotes5• 4 years ago

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.

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.)

TimTheTinker3• 4 years ago

Must have been for degree 4 or below.

acer4666• 4 years ago

And how does "no closed form solution" imply "no solution"?

charlieyu1• 4 years ago

I figure out the formula of the number of diagonals of n-sided polygon within 15 minutes when I was 10. I thought I would be the greatest mathematician ever. Gauss figured out the formula of 1+2+3+...+n at a similar age and the difficulty was similar, right?

I have never been so wrong haha

eyelidlessness• 4 years ago

I was by no means this kid. But I remember many times I’d passively listened to the lesson of the day, then get called on because I seemed (was) distracted, and offered a simpler solution. Usually got a “we’ll cover that in future chapters” with very little attempt to hide the frustration.

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.

geebee• 4 years ago

I'm going to be a little obnoxious here. Did he really implement a factoring program, or did he approximate the solution to a high precision numerically? Both would be impressive under the circumstances of course. But if he did it numerically, he still might want to pay attention to how to derive the quadratic equation using partial fractions. It's actually pretty clever.

dgs_sgd• 4 years ago

Congratulations to him! That's an amazing accomplishment. There's a lesson in this line

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.

skyhvr• 4 years ago

To your point, one of the more fascinating details about John Nash from Nassar’s book ‘A Beautiful Mind’ was that this very trait set him apart from a number of his contemporaries. He was capable of focusing on problems for what appeared to be quite a bit longer than most, systematically breaking down the barriers that would’ve turned another person toward some other pursuit.

dekhn5• 4 years ago

I've known quite a few families where both parents are scientists or software engineers or quantitative engineers.

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.

misterprime2• 4 years ago

Agreed, but this brings up the topic of nature vs. nurture, and perhaps even the previously (~90 years ago now?) popular concept of eugenics.

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.

nominusllc• 4 years ago

I wish this could be discussed more openly. I feel the same way.

feoren2• 4 years ago

I think society settled on a response of "yeah, but we shouldn't actively do anything about it as a group."

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.

dekhn5• 4 years ago

This isn't eugenics. People are allowed to choose their mates for their phenotypes, and we don't sterilize (except in very rare, extreme circumstances, with legal approvals) people who don't have specific phenotypes.

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.

Quekid52• 4 years ago

Fundamentally, even ignoring the ethical problems, eugenics doesn't work. The interactions between nature and nurture, genetics, epigenetics, etc. are waaaay to complex (and even multi-generational) to boil down to a simple decision to allow/deny birth.

strikelaserclaw4• 4 years ago

i mean, we should be able to get to a future when everyone can enjoy in the bounty produced by technology, it is arrogant to think humans with their limited wisdom can determine who gets to breed and who doesn't

schwartzworld• 4 years ago

I don't think that's entirely Fair. Anything you learned at a young age you're going to have more potential in. If you were introduced to X as a kid, you have years of doing or practicing X that someone else will never have, and those years come when your brain is still developing.

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.

strikelaserclaw4• 4 years ago

part of this is they are probably wired for it, and part of it is they are immersed in it from a young age. Probably don't need to worry though, a lot of geniuses come from left field where they grow their creativity in isolation.

hi41• 4 years ago

Couple of years back there was an article that talked about higher incidences of autism in kids with such parents.

NoImmatureAdHom2• 4 years ago

That's because intelligence is ~80% heritable! Or rather, ~80% of the variance in intelligence is explained by things you inherit.

tylerhou• 4 years ago

(Note: "inheritable" here is not exclusive to inherited genetics; it also refers to upbringing.)

djmips• 4 years ago

In another area - I knew a guy and his parents were very good with finance and business and he was way ahead of everyone else right out of high school. He did go on to become very successful financially. And me, well I should have done better! (Oh yeah, don't compare yourself. Run your own race. hmmm)

kristopolous2• 4 years ago

success and progress is about favorable contexts and resources

goy• 4 years ago

Nobody is talking about the proof, but about the fact that the person who produced it was younger than them and they are trying to explain the achievement by innate abilities or parent influence. The fact is just that most of people don't want to do something like that. They just want to be someone who do things like that.

YeBanKo• 4 years ago

The article talks about the proof and how he came up with it as much as it talks about his age.

polynomial• 4 years ago

Of course that's just the first (coarse) filter. Once you've selected for those who actually want to solve the problem (and not just "be" the one who solves it) there is still the the incredibly fine filter of who actually has the ability- that confluence of nature/nurture that is itself such a subject of debate.

120bits• 4 years ago

Congratulations! This is a great achievement at his age. Maybe a Fields medal next?

mkoryak• 4 years ago

My daughter is in 3rd grade and is bored with her math lessons, so I started to do my own nightly math lessons. Long division was a big hit, but what I really want is to find some accessable number theory material, preferably about interesting patterns. Any recommendations?

Probably something in the 10-15 age range, or 8-10 and originally in Russian

kotlin2• 4 years ago

I remember something a math teacher gave me in middle school as extra work. It was a packet of patterns and it gave you the first few items in the sequence and you needed to write a formula that expresses the value for the nth term. Something like this: https://corbettmaths.com/wp-content/uploads/2018/12/nth-term...

ducttapecrown• 4 years ago

https://artofproblemsolving.com/

quacked• 4 years ago

My best friend was a child prodigy at math and his parents started him on "The Number Devil" which I believe is number theory

pfdietz2• 4 years ago

This guy has Sitzfleisch, a very useful thing for a mathematician to have.

Der_Einzige• 4 years ago

Do Germans really say that people with grit have a lot of "butt" because that's what its looking like here!

pfdietz2• 4 years ago

Yes, that's the other meaning of the word. :)

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.)

cod1r• 4 years ago

Weird how this came out on the day I bomb my probability test. T_T

kristopolous2• 4 years ago

Quanta is doing really solid science journalism these days. They have a number of fantastic podcasts as well.

It's bankrolled 100% by this generous billionaire https://en.wikipedia.org/wiki/Jim_Simons_(mathematician) and I hope he continues to fund it

Teenager solves stubborn riddle about prime number look-alikes

Related Stories