Partitions - Numberphile
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8: 37 No other such identities known where the coefficient of k is a prime, but similar identities have been known where the coefficient of k is a power of 5, 7 or 11. 17303 = 11-3 - 13. :) 206839 = 17 - 23-3. 1977147619 = 19 - 101-4. - Interesting that this is the first appearance of 23 and 101.
Date: 2022-04-08
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Comments and reviews: 9
xenolalia
At 10: 04 Dr. Grimes says of the classic Hardy-Ramanujan series approximation to the partition function: -in fact it becomes equal [as you include more terms of the series]. - However, this is incorrect, as the Hardy-Ramanujan formula is only an asymptotic approximation (i. e, the value of the kth partial sum converges to p(n) only as n --> inf for fixed k, but -not- as k --> inf for fixed n. It was actually Radamacher who, in 1937 (some twenty years after the publication of H. and R. 's original result, was able to modify their formula to make it absolutely convergent.
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At 10: 04 Dr. Grimes says of the classic Hardy-Ramanujan series approximation to the partition function: -in fact it becomes equal [as you include more terms of the series]. - However, this is incorrect, as the Hardy-Ramanujan formula is only an asymptotic approximation (i. e, the value of the kth partial sum converges to p(n) only as n --> inf for fixed k, but -not- as k --> inf for fixed n. It was actually Radamacher who, in 1937 (some twenty years after the publication of H. and R. 's original result, was able to modify their formula to make it absolutely convergent.
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ade
such a fascinating subject. I dont know much about maths, but when I read about Ramanujan I remember thinking this guy was one of, if not the most naturally gifted mathematicians in History. Had very little formal training, and so poor, he couldn't even afford notebooks, so had to use slates to do his calculations on, what mathematical marvels were lost on those slates, though he did of course, keep his best ideas in his three notebooks. Those slates are possibly in landfill somewhere.
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such a fascinating subject. I dont know much about maths, but when I read about Ramanujan I remember thinking this guy was one of, if not the most naturally gifted mathematicians in History. Had very little formal training, and so poor, he couldn't even afford notebooks, so had to use slates to do his calculations on, what mathematical marvels were lost on those slates, though he did of course, keep his best ideas in his three notebooks. Those slates are possibly in landfill somewhere.
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education
Probably my favourite story of Ramanujan is when G. H. Hardy went to see him, and I'll let Hardy tell the story: -I remember once going to see him when he was 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 unfavourable 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.
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Probably my favourite story of Ramanujan is when G. H. Hardy went to see him, and I'll let Hardy tell the story: -I remember once going to see him when he was 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 unfavourable 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.
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Shiva
Ramanujan said it's Goddess who in his dreams used to continuously write those formulas or results (without proof) and when he woke up he used to write those.
But since he does not have mathematical training he gave those formulas to Hardy and Littlewood to prove it.
What a poor fellow those Hardy and Littlewood were. They probably went mad and half lunatic with those formulas which are direct revelation from Goddess through Ramanujan!
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Ramanujan said it's Goddess who in his dreams used to continuously write those formulas or results (without proof) and when he woke up he used to write those.
But since he does not have mathematical training he gave those formulas to Hardy and Littlewood to prove it.
What a poor fellow those Hardy and Littlewood were. They probably went mad and half lunatic with those formulas which are direct revelation from Goddess through Ramanujan!
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Cold_Blooded_
intuitively, i would have thought, that the pattern would look like this: 5 -> 7 is +2, 7 -> 11 is+4 and then either it would be 11 -> 17, because we always add 2 or it would be 11 -> 19 because we always multiple by 2 of what we add, so the next one in pattern should look like: P(19k +7. Thats what i would think, but of course it does not works, but i still wanted to share this; )
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intuitively, i would have thought, that the pattern would look like this: 5 -> 7 is +2, 7 -> 11 is+4 and then either it would be 11 -> 17, because we always add 2 or it would be 11 -> 19 because we always multiple by 2 of what we add, so the next one in pattern should look like: P(19k +7. Thats what i would think, but of course it does not works, but i still wanted to share this; )
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Grandpawns
I haven't tried solving it yet but it seems a simple combinatorics problem
imagine a number of 1's with + signs between them
it's just a question of in how many ways can we group them
it should be fairly simple
but way smarter people have tried and failed so what's the difficult part
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I haven't tried solving it yet but it seems a simple combinatorics problem
imagine a number of 1's with + signs between them
it's just a question of in how many ways can we group them
it should be fairly simple
but way smarter people have tried and failed so what's the difficult part
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Simon
I said it once I'll say it and every video it's mentioned energy is never conserved
Everyone talks about the heat death of the universe and that is a scientific proven fact that if no one interferes the heat death of the universe will occur. And there is no conservation in that process
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I said it once I'll say it and every video it's mentioned energy is never conserved
Everyone talks about the heat death of the universe and that is a scientific proven fact that if no one interferes the heat death of the universe will occur. And there is no conservation in that process
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DarkRedman31
I disapponted that there is the newer formula in the video but not explanation about what is x and what is k, because we only want the number of possible permuations, so why two number and which one is the number of blocs!
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I disapponted that there is the newer formula in the video but not explanation about what is x and what is k, because we only want the number of possible permuations, so why two number and which one is the number of blocs!
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J. Azz.
I simply cannot understand Ramanujan's forumlas. They seem like basically a bunch of random numbers stuck together that somehow work. Even more because they never seem to be equal, but approximations.
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I simply cannot understand Ramanujan's forumlas. They seem like basically a bunch of random numbers stuck together that somehow work. Even more because they never seem to be equal, but approximations.
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