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Classic Math Puzzle Cracked
An anonymous reader writes "This is cool - if mind-bending. A century ago, a self-taught math genius from India noticed some patterns in how numbers can be created by adding other numbers. Now a grad student has finished the job showing that the patterns apply to all prime numbers, not just some. There's more on the Indian math guy here."
It is interesting that the New Scientist article basically attributes the idea of studying number partitions to Ramanujan, yet the linked article on him mentions that Euler had studied the problem before, and given a partial solution...
Repton.
They say that only an experienced wizard can do the tengu shuffle.
GH Hardy (he wrote A Mathematician's Apology) speaking of Ramanujan:
I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
(London 1940).
"The more you know, the less sure you are." - Voltaire
While this allows one to calculate p(n) recursively, it doesn't lead to an explicit formula. Hardy and Ramanujan came up with such a formula (though they only proved it works asymptotically; Rademacher proved it gives the exact value of p(n)).
The value was already known exactly. There was already a formula which could calculate it easily, and it had solid theoretical and empirical indications it was correct. Now we have a stringent proof. Big whoop.
Not to undermine its importance *in mathematics*. But I think we can safely say the world won't exactly change over this proof.
Kjella
Live today, because you never know what tomorrow brings
I suppose Evariste Galois was just some guy who flunked his entrance exams and was killed in a duel when he was 20? What would you have to say about Paul Erdos?
(snicker)
You're putting Linus in the same crowd as Einstein and claiming Ramanujan is "some grad student"?
Keep'em coming. I needed a good laugh. We'll see how long it takes for my ribs to hurt. You have a long way to go to top the time in college when we crumbled Gaines Burger dog food and put it into the salad bar, but with some work and a good mentor, you too, might be funny.
Wait, I almost forgot. Are you trolling?
bwahahahahahahahahahahahaha
No.
Compression algorithms map one huge number (consider an entire file as one huge number) to another. They "work" because most huge numbers of interest in a given domain aren't valid; random ASCII is gibberish, not English, so we remap that "random" looking stuff to stuff of more interest. This allows us to pack the interesting things much more tightly into the small numbers.
But for every number we shorten, we must also lengthen a number. Real-world algorithms do clever things to minimize the real-world impact of this fact, so you don't see it, but it's obvious if you think about it. If you have a sequence "1 2 3 4 5 6 7 8 9 10" which maps back to 1-10, for every number you pull down (move 8 -> 2), another number moves up.
No matter what you do, you can't create a magical compression algorithm that can be the "DNA" of all other numbers. You didn't say this directly, but a lot of people have this idea floating around in their head and I sort of "smell" it in your post.
(Proof: Suppose you have a compression algorithm that always shortens a number, and the corresponding decryption function. (Note we don't assume anything about the nature of the algorithm other than the compression, so it applies to all such algorithms, no matter how fancy the math.) Of the binary numbers 00, 01, 10, 11, each is therefore shortened to 1 bit. But there are only two possibilities for that one bit, and it has to cover 4 numbers. This is not possible for a decompression function by definition of "function". Therefore, contradiction, and there is no such compression algorithm.
I left the terminology a little fuzzy to try to prevent Math Overload; mathematicians should be able to fill in the blanks fairly easily.)
That said, the man was brilliant. It is only too bad that he died before he could do any more than he did. He had the potential to make breakthroughs in the same way that Newton, Einstein, etc. did. If only Hardy had let him continue to work in India...(Many people attribute his death to the unfamiliar climate of England. I know that he died of TB, but it is likely that he wouldn't have caught it had he remained in India.)
All your
Excuse me for not being a Linus fanboy, but Linus is NOT in the same league as Einstein. I'm surprised they're mentioned in the same sentence. When Linus dies, nobody will be falling over themselves to dissect his brain.
Maybe, but after looking at the link about SR's origins, the "Indian Math Guy" does seem sort of appropos. He certainly was that until Hardy essentially recognized the guy's talents.
If there was a world-class Cricket or Squash player, he/she'd probably be known as "that American Cricket Batter", etc.
Part of that is perception of reality. In this case, at the time, India was a British Colony. As such, expectations of Indians by British society were...low.
Just like the expectation of an American being a world-class cricket player (who is not an expatriate from a traditional 'cricket' country).
Just look at the angst in the US about long distance running. The US hasn't had a world-class marathoner or 10K runner since the mid-80's.
And yesterday we had "a major Australian newspaper" omitting to mention it was the "Melbourne Age". It seems that foreigners aren't worthy of having names, so it's just a waste of space to use them. But "they" really rubbed it in with this one, mentioning Ramanujan's nationality twice and still avoiding the name -- though perhaps it would have been more insulting if they'd tried to use it, considering the quality of spelling here.
"If only Hardy had let him continue to work in India"
IMO:
He could not have worked in India. He needed a lot of personal tutoring and contact with first-rate mathematicians, and there haven't been many mathemeticians as first-rate as G. H. Hardy.
Whether the early death was worth (to the world or to Ramanujan) the growth (to math, to Ramanujan, and to Hardy) that came from the Ramanujan-Hardy collaboration, I don't know.
Exam 4/C again. Maybe I'll do better this time.
India has a very long history of mathematics. eg. Pythagoras theorom was proven in India long before Pythagoras was even born.
Engineering is the art of compromise.
This sounds like it was posted by a Seventeen reader. A new low for /.
Those who would like to learn more about the "Indian math guy" should read the excellent biography, The Man Who Knew Infinity".
While I did quite well in prep school in mathematics, my love for mathematical symmetry morphed to the study of music theory in college (the two are not as dissimilar as most would think).
That's certainly true.
How's this for a strange coincidence.
It would be tough to name the greatest mathematician, or the greatest composer of all time since there is a great deal of subjectivity to that.
However, I believe that it's pretty generally accepted that the greatest musical dynasty was the Bach family and the greatest mathematical dynasty was the Bernoulli family.
There were at least 3 generations of some of the greats in their respective fields.
They lived at the same time, and within 100 miles of each other.
I'm not saying it means anything, but it truly amazed me the first time I learned about it.
Einstein was very smart.
I wouldn't want to put him down.
But I agree that Ramanujan was a phenomenon. He was so completely different from any of his contemporary mathematicians that there is really no comparison.
He was discovered by the west when he sent a manuscript to Hardy, a famous English mathematician. Hardy almost discarded it, since much of it was stuff he had seen before (though Ramanujan had rediscovered it independently), but it also contained 120 thereoms no one but Ramanujan had ever seen before.
Later, when he came to England, Ramanujan filled notebooks with thousands of theorems, though not, apparently with proofs. I think proving Ramanujan's thereoms is still a major occupation of academia.
Interestingly, there is a similar story involving Einstein. Bose, who was an unknown Indian physics instructor, sent an unsolicited manuscript to Einstein which eventually led to the theory of Bose statistics, or Bose-Einstein statistics and the Bose condensate.
Crackpots from all over the world were sending Einstein manuscripts, and Bose's manuscript looked a lot like one of these. But Einstein read it anyway, and saw that Bose's ideas had merit. Ultimately, it seemed that Bose only had the one really good idea in him, and after collaborating with Einstein on the one paper, he went back to India and continued teaching. Apparently he was an especially good teacher.
MM
By including this sig, the copyright holders of this work or collection unreservedly place it in the public domain.
Few would remember a name from a distant culture. But many would remember that there was a math genius from India in the early 1900's if they had heard his story once.
There was another genius like this, only he was a musical genius. There was an African-American slave in the mid-1800's who could play nearly anything on the piano after hearing it once or twice. He was a 'field slave', not a 'house slave'. He used to sneak up to the plantation manor house and listen to visiting musicians play Bach and Mozart on the piano. He was caught one night playing Bach on piano in the manor house and only escaped being whipped to death by his unbelievable talent. He also had the ability to sit down at the piano and play any chord that someone else had just played. He could do by ear.
His 'master', the plantation owner, took him on concert tours around the US, even to the North where this black genius was not a legally-owned slave and would have been able to receive politcal asylum and freedom. But he always returned to the plantation with the 'master', as he was illiterate and uncomfortable among the northern wealthy gentry.
I know that this guy existed; he was a genius whose type of talent appears only in one of ten million people, but I have no idea what his name was. Maybe some Slashdotters who are seriously into African-American musical history could let us know.
Now start to get your head around this... what if numbers are just, well, *human* constructs. As hard as that is to get your head around, think how easily that little concept completely changes your view of things.
If numbers are human constructs and nothing "inherant" in the universe, then the patterns that we find are not that unexpected. Humans are pattern hunting machines.
"Your superior intellect is no match for our puny weapons!"
I assume these numbers are added to numbers to create (astonishingly) numbers. And this operation can even be applied to all prime numbers! This is really mindbending and puzzling and probably innovative too. Is this method patented yet? Hey, i got a great idea: let's use the "+" sign for this operation, something like "+(number1,number2)", i think i'll patent that.
Maybe that anonymous reader should've freed himself from the mindbended state briefly and taken the few extra seconds to specify "numbers" for the benefit of the readers.
"By the way if anyone here is in advertising or marketing... kill yourself." -- Bill Hicks
You are correct, I'm terrible with names.
"The term was coined by René Descartes in 1637 in his La Géométrie and was meant to be derogatory: obviously, such numbers were thought not to exist."
This statement does not give him enough credit.
The word has been misstranslated.
Imaginaire
Of the mind, or
Image-less
Invisible
Vision less
He described them this way because they could not be plotted using his cartesian coordinate system, not because they obviously didn't exist. He used them, and new better that that.
If voting were effective, it would be illegal by now.
Your explanation is good, but I think maybe too complicated for the average punter. I tried to write the simplest one I could and came up with the following. Any comments appreciated (especially if it makes it simpler; the only vaguely technical word I've used in it is "compresssion" and since that's the topic, I think that fair):
1 10 00000000000000000000000000000000000000000000111111 111111111111111111111111111
Why you can't keep compressing a computer file and why no system of compression can compress every file.
The most important thing to remember here is that computer files are just numbers. BIG numbers, right enough, but numbers none the less. For example:
11111111111111111111111111111111111111111111111
or
10110001 00101101 10100001
could be computer files (pretty short ones, but they're just examples). Now, obviously, if you're compressing a file, you're representing a big file by a smaller one. For instance, we could represent the first number by the second one.
Consider if you had to represent all the numbers up to 100 with just 1, 2, 3, 4 and 5. Not "51" or "43", just those five numbers. Well, of course, you can't. You could have 1 == 67 and 2 == 83 and 3 == 98 and 4 == 55 and 5 == 12 but then you're out of numbers. You can only represent five different ones.
So, that's why you can't have a universal compression scheme, and why you can't keep compressing a compressed file: because there are more big numbers than there are small numbers!
Igor Presnyakov stole my hat
Mir Sultan Khan arrived in England in 1929 as manservant to an Indian Maharaja, and immediately took the European chess world by storm (the Wikipedia article compares him to Morphy). He convincingly defeated all the great players of that era -- Alekhine, Capablanca, Euwe, Rubenstein, more, but when the American master Reuben Fine visited the maharaja's digs in London, Khan was the waiter who served the meal. In 1933, the maharaja left England and Khan was taken back to India: no more tournament chess for him.
His story is not the same as the story of Blind Tom, in spite of cetain similarities. There is no indication that Khan's owner/employer exploited those remarkable talents, and the talents were in fact measurably remarkable. In the case of Blind Tom, one is tempted to think of S. Johnson's remark: "Sir, a woman's preaching is like a dog's walking on his hind legs. It is not done well; but you are surprised to find it done at all." [from Boswell's Life of Johnson]
ZERO is from India (the back office,3rd world country etc.) I think its time to come up with new idea to build only using ONE. Let ZERO live alone. --
Education should be so revolutionized as to answer the wants of the poorest villager, instead of answering those of an imperial exploiter. - Mohandas Karamchand Gandhi "Mahatma"
OK, so I'm an idiot in maths, and I've read about prime numbers and cryptography and how predicting prime numbers can help crack encrypted material, so is this development of any significance with cryptography?
You're referring to the main "unamed" character in James Weldon Johnson's book, The Autobiography of an Ex-Colored Man. This book is fiction and AFAIK, the character was completely made-up. I could be wrong.