Interesting… but how large are we talking?
Can’t help you there (and after searching more, I may be talking rubbish).
I think I remember reading about it on some mailing list, (perhaps FOM?, though I cannot find anything relevant under a search for “large cardinals” there, now).
Though there’s this paper, which I can’t seem to download, which seems relevant.
'bout like this.
Ogre writes:
> Redacted for my own sanity. Suffice to say that the math snobbery in this
> thread is pretty disgusting. I wonder how many of the pooh-poohing
> mathematicians could have done that at 16.
I’m sorry, Ogre, but you don’t know what you’re talking about here. First, let me establish my credentials as someone who does know what they’re talking about. In my life I have met many absolutely first-class mathematicians. I’ve known a Fields medalist. I’ve known someone who was a Putnam Fellow all four years in college, and I’ve known two other people who were Putnam Fellows three years in a row. I’ve known someone who was probably the youngest person ever to become an assistant professor of mathematics and someone else who was possibly the youngest person ever to become a full professor of mathematics. In college, grad school, and my working life I’ve known many other mathematicians who were nearly as good. I was in the Ross Mathematics Program in the summer after high school, and the average quality of the students in that program was possibly better than the average math students at a top college, the math students at a top grad school, or the mathematicians I’ve known in my working life. (Look up the Fields Medal, the William Lowell Putnam competition, and the Ross Mathematics Program if you haven’t heard of them.)
I’m not saying this to show off. By the standards of these people, I’m a failure. I scraped through to my master’s in math and have worked the past twenty-seven and a half years as a mathematician, but I’ve also barely scraped by at that too. I’m only pointing this out because someone is going to claim that I don’t know what a really first-class mathematician is. I do, and you don’t know how one would be discovered, Ogre.
The notion that great mathematicians are nearly always prodigies is greatly exaggerated. They are sometimes, but mostly they are people who do reasonably well at each educational level (elementary school, high school, college, and grad school). In general, it’s not until the end of grad school that first-rate mathematicians pull out in front of merely good ones by proving important new theorems.
Some mathematicians publish their first papers in college, but that’s rather rare, even among first-class mathematicians. Mostly they don’t start publishing until grad school. I don’t know any cases offhand of someone publishing a mathematical paper in high school, but I suppose a few such cases exist.
What then would a mathematics professor do if he received a paper from a high school student who claimed to have proved an important theorem? Well, his first thought would be that this paper came from one of those nut cases who constantly annoy mathematicians with their claims to have proved some ridiculous theorem. Sometimes they claim to have proved something like how to square a circle or how to trisect an angle (using only compass and straightedge). These theorems are well known to be false. Sometimes these nut cases claim to have proved some theorem that makes no sense whatsoever. I suspect that most claims of proofs received from amateurs are of this sort.
The mathematics professor, having reluctantly decided to read this paper, would be happy to discover that it wasn’t a nut theorem but something that actually made sense. He would next be expecting that there were be some mistake in it that invalidated the proof. Having finished reading the paper, he would be happy to discover that it was a correct proof of a theorem. In this case, the professor would also quickly find out that it was a previously known result. Since the person who sent it in was sixteen years old, he might send him a nice letter congratulating him on the result and mentioning that he might consider applying to his university when it comes time. He would also tell himself that, for once, he had read an amateur’s paper that wasn’t a complete waste of his time. What he would never do in any circumstances is offer the person a place at his university. He couldn’t do it anyway, nor could the admissions office. (Among other things, if someone sent in a good but non-original proof of a theorem, one would have to think about the possibility that he had merely copied it from a book.)
People have the bizarre notion that mathematicians are constantly looking for brilliant prodigies to appear on the scene by proving some wildly new and important theorem. Mathematicians don’t expect to ever find any such thing. In general, prodigies aren’t nearly as important in mathematics as most people seem to think they are. In general, mathematical prodigies do pretty well, but not that much better than mathematicians who go through school at the normal rate. (Here I am defining a prodigy as someone who goes through their educational career much faster than other people.) This is true in other fields too. Occasionally a prodigy burns out. Usually that means that they lose interest in the field and move on to something else. Most of the time, even though they finish their education much faster than others, some of the best of the people who do their education on the normal schedule will do even better at their careers than the prodigies. On rare occasions a prodigy does turn out to be the one who does best at his career, but that’s not nearly as common as you might think.
As a rule then, prodigies do well but not brilliantly well. That’s why there’s no way to say from the fact that this sixteen-year-old created a (non-original) way to generate the Bernoulli numbers that he’s going to do fantastically well in the rest of his life. The chances are that he will do quite well in the future, but that’s about the best one could say.
So the Bernoulli numbers are the coefficients of X expressed as a series as one proceeds through the formulae for calculating the sums of the whole numbers to the power of zero, power of one, two, three, etc.
Cool.
Is there any name for, or significance to, the series of co-efficients of X**2? Or higher? And is there a meta formula (similar to the one discovered by the Iraqi kid, but at a higher order of generalisation) for generating not just Bernoulli numbers, but all the numbers of Bernoulli-like series I just described?
Don’t bust a gut. I’m just curious.
You can determine all the rest of the co-efficients from the Bernoulli numbers as well; in the formula for summing p-th powers, the coefficient of X[sup]k+1[/sup] will be the (p-k)-th Bernoulli number times p!/((k+1)! * (p-k)!).
Thanks! Very cool indeed. I vaguely thought that something like that might be happening when I saw that the co-efficient for X**2 in the formula for adding up the squares was 1/2, in Post #34. But that was just a guess, based on no serious mathematical knowledge whatever. I do like the way your expression of the formula for p-th powers includes so many !s that you look positively excited about the result. At school, the nomenclature we used was a little L-shape written sort of around the number. All mathematicians are over it, but because of my schooling I still enjoy seeing the ! convention used for factorials.
Where did you learn to use an L around the number rather than a ! after it for the factorial? I’ve never seen anything like that before, and I can’t find any mention of it on a quick Internet search.
What a long-winded, completely nonresponsive response. None of it explains why you’re aggressively “meh”-ing the accomplishments of a 16-year old kid with the drive and talent to do higher level mathematics. I never used the word “prodigy”. That was all you. I have ZERO illusions that the world of math is like “Good Will Hunting”, and I’ll thank you to speak to me like an adult, and a scientist (who had lunch with a Nobel Prize winner a few weeks ago, thanks very much), and not like a rube who gets his information from Dr. Phil.
In conclusion: 16. Significant desire and drive. Should be nurtured, not dismissed with a hand-waving “you’re work wasn’t entirely original anyway.”
I wasn’t downplaying this sixteen-year-old accomplishments at all. He will probably turn out to be a pretty good mathematician, if that’s what he wants to do. (And it’s quite possible that he will want to do something else entirely, which is just fine if that’s what he wants.) I was responding the attitude in the OP, which is that it’s possible to tell from what someone does at 16 how well they will do for the rest of their life. So he found out how to generate the Bernoulli numbers, although it wasn’t an original discovery. That’s great, and he should be encouraged, but it doesn’t remotely mean that he will win a Nobel Prize (or a Fields Medal, which is closer to what’s relevant). You’re the one who’s misreading other posters by claiming that we were pooh-poohing his accomplishments.
Well, excuse the tar out of me. I know for myself, advanced mathematics is a foreign language that I will never be able to learn. I found it impressive that a young boy seemed to do what no one else could. (I know now that wasn’t the case, but still, he grasped something I’ll never get.) I tend to be impressed when people do remarkable things, whether it’s this kid or Susan Boyle wowing BGT judges or firefighters rescuing people from burning buildings.
I for one want to thank the folks who’ve been working on trying to bring the explanation to a moderate laymen level.
My bachelors was in Computer Science with a minor in math. I remember most of Calculus and can recognize the various classes of differential equations I used to be able to solve. I did a lot of finite math, number theory, and topology, at least at the bachelor’s level. Nowadays I can remember the vocabulary words and have a rough outline of their meaning. Comprehending abstract ideas is not my weak spot.
I don’t glaze over at the word “polynomial”, unlike some of the folks bleating for an explanation. I also got my butt chewed in another thread recently where I was the expert and some layfolks didn’t like the answers I provided. So I sympathize with the challenges inherent in picking the right level of simplification and trying to make a coherent story which is both fundamentally accurate and not so strewn with caveats as to be incomprehensible.
Like a lot of things in discrete math, it sounds like the Bernoullis are just a little too far out to feed to the true laymen. They’re foundational, but it’d take a 2 week intro to bring enough of the rest of the foundation into focus for layfolks to get the picture.
Thanks again for trying.
Well-established private high school in Australia in the 70s. (I say well-established, because “private school” in America may sometimes carry hints of nutjobbery, I understand, which was definitely not the case here.) I assume the state schools were doing it as well, because it is likely that the curricula were very similar, but I don’t know that for sure. I don’t know where the L convention came from. I knew about the ! convention then as an alternative to what we were doing, and we were taught not to use it, but never told why beyond the obvious need for consistency. I had thought the L convention was English, in the spirit of the different uses of the word billion, etc, which was where much of our cultural baggage at the time originated.
Or maybe someone somewhere just didn’t like the look of perpetual astonishment that equations with the ! bear. 
Belated edit to my post two above this one …
Yes, that’s it exactly. And as I noted above, the fact the experts have had a hard time doing even that indicates they’re perhaps just a bit too obscure for the audience here. But thanks for playing so far.
It was me who said the word polynomial meant nothing to me. However, I’ve been nothing but polite to the people who’ve tried to provide explanations and have tried to rephrase the explanations in my own words to see if I’d understood it correctly. There was no ‘bleating for explanations’ on my part. If someone doesn’t understand an explanation, are they supposed to just pretend they do?
I still don’t know if my rephrased explanations were correct or not, because no-one seems willing to tell me if they are, but we all have better things to do with our time, I guess.
This is what I was just about to say. No-one has been “bleating for an explanation”. All the Non-Maths people in the thread have been very polite and I believe we’ve all thanked the contributors for trying to explain something that appears to be Just Too Complicated To Explain To The Layperson.
Even “It’s a type of equation that’s really, really hard to solve and not many people have managed” would suffice for probably 95% of the people who wandered in here and said “What’s a Bernoulli number and why is this news?”, I think.
If something is JTCTETTLP, then I don’t see a problem with someone politely saying “Listen, this is really, really, REALLY complicated stuff that is unlikely to make sense to anyone without at least an undergraduate degree in a Mathematics related field. There’s no way to provide any sort of meaningful explanation to someone without that basic groundwork understanding, unfortunately. Sorry.”
There’s probably as much work involved in finding a tactful way to say that without coming across as an arrogant dick as there is in solving a Bernoulli equation, though. 
I am a mathematics graduate student in algebraic number theory studying class field theory. I’ve come across Bernoulli numbers in Kummer’s work on the proof of Fermat’s Last Theorem for regular primes. Both Indistinguishable and Wendell Wagner have provided excellent explanations of what Bernoulli numbers are, and I had hoped to give an enlightening explanation of how they are useful in one specific example, but I fear I will not be nearly as successful as the other two distinguished contributors.
Bernoulli numbers and Bernoulli polynomials play an important role in Kummer’s proof that there exist infinitely many irregular primes.
That’s it. That’s all I’ve got for you. If we try to get more specific, we will quickly get bogged down in technicalities, and at least an elementary grasp of algebraic number theory and ideal theory will be necessary to continue. The previous paragraph merits elaboration, but I do not see how it is possible to explain the role these objects play to a layman on an internet message board.
I hope I have not offended anyone at all. I don’t like the math vs. non-math quality of this thread. Number theory is a topic I find interesting, and I love to share my interest with anyone willing to listen. I just don’t think in this instance I am able to share it particularly well.
And finally, Noel Prosequi, the little L around the numbers you mention, could that possibly be the floor function you are remembering?
Just for the record, I found your explanation of Bernoulli numbers very clear, Wendell Wagner, even with a few typos. Interesting stuff.
In answer to the question “Why are Bernoulli numbers important? Even given that they tell us about the sum of the n-th powers of integers, why is that important?”, the answer is that they turn up in lots of surprisingly different places in various areas of mathematics. Unfortunately, it would take us a lot of time to explain all the various places that they turn up. Not having made a serious study of Bernoulli numbers myself, I can’t even tell you much about this subject in any case.
The problem is that it’s kind of like saying that looking for oil here was useless because you found none, and that from now on you should only look for oil where you know it exists.
Many aspects of advancement of the understanding of pure math may never be any use for anything. Some are. Knowing which are going to turn out to be which would require a greater knowledge of the subject matter than we have. And we can only get that greater knowledge by chasing down every lead and seeing where they end up.