5 ms·

An elliptic curve of rank ≥ 30

104 points by robinhouston 1mo ago

zenburnmyface 1mo ago
Want to understand this more? Read the triplet of books by Ash and Gross, specially Elliptical Tales
- nhatcher 1mo ago
  Hehe, I really like them. And I always recommend those. It's not always that you have some nice books on deep concepts in mathematics for the "general" public
dwrensha 1mo ago
I'm the maintainer of the linked website. This morning, a mysterious user named "ranksunbounded" submitted the linked curve, which has rank at least thirty. This breaks the previous record of 29 found by Elkies and Klagsbrun in 2024. Nobody knows whether it is possible to achieve arbitrarily high ranks.
- impendia 1mo ago
  I'm a research mathematician, and to me the biggest surprise is that the user didn't identify themselves or (yet) try to claim credit for their work. Within academia, it is to your definite advantage to have your name associated with breakthroughs, for obvious reasons. Outside academia I don't have firsthand experience, but I'd have to imagine the same holds.
  - bdamm 1mo ago
    If this work has an implication for cryptography they might want to be anonymous for any number of reasons.
    - charlieyu1 1mo ago
      Or a random amateur who got lucky with AI. This happened to someone I know and he doesn’t even know how to publish
  - shshshhss 1mo ago
    It is also a sign of wisdom and maturity to not care so much, especially outside academia. I know your reaction is the normal one these days, but I’d like to normalize the ones that don’t care about society petting them on the head and saying atta boy. They just deliver the work and retreat into the darkness. I find it highly inspiring. The alternative.. not so much.
    - voakbasda 1mo ago
      Agreed. This is what real research looks like. Not publish or perish. Not going after a bunch of grants. Just doing the work and getting results. Sadly this is this antithesis of modern academia.
  - impendia 1mo ago
    Update: the linked page has now been updated to say: > it was Claude, with Levent Alpöge and Ava Howell! Alpöge is the same guy who prompted Claude for a proof of the Jacobian conjecture.
    - fifdspence 1mo ago
      He may have a penchant for unusual announcements. Making an "ranksunbounded" user and posting a 30 today, 31 tomorrow, and 32 the next wouldn't shock me.
  - TomatoCo 1mo ago
    You might appreciate the first author on this paper, https://oeis.org/A180632/a180632.pdf https://oeis.org/A180632/a180632.pdf https://en.wikipedia.org/wiki/Superpermutation#Lower_bounds,_or_the_Haruhi_problem https://en.wikipedia.org/wiki/Superpermutation#Lower_bounds,...
    - robinhouston 1mo ago
      Funny coincidence: I (who submitted this story) am the second author.
      - TomatoCo 1mo ago
        I hadn't read carefully enough to recognize that! That's a wacky coincidence that I love. Thank you for your writing.
- dwrensha 1mo ago
  ranksunbounded has added a comment on the curve: "it was Claude, with Levent Alpöge and Ava Howell!"
  - bmenrigh 1mo ago
    I just don't understand how Alpöge can get Claude to construct so many complex mathematical objects. He must already have some deep insight into the structure these things are likely to take. Naive prompting, or naive search would not have found pretty much any of the object he's found in the past month.
    - Resident1324 1mo ago
      He has an AB in Math from Harvard with a senior thesis "The average elliptic curve has few integral points", and a PhD in Math from Princeton with thesis "Points on Curves", so it's fair to say he has some deep insight. (https://alpo.ge/cv.pdf https://alpo.ge/cv.pdf)
      - bmenrigh 1mo ago
        Okay this result may well be in his expertise wheelhouse but what about the recent smooth Carathéodory conjecture disproof? Or the Jacobian, or the Hadamard matrices? It seems to span a pretty wide range. I know others are getting good results too, but his really stand out.
    - ailunicguy 1mo ago
      It wouldn't surprise me if he is given access to a more advanced model or more compute, since he is an Anthropic employee. Add to that the fact that this makes great PR and that he has not shown any of his prompts, it's more than likely in my opinion that this is the case.
great_psy 1mo ago
Can we have a tldr of the implications of this for the uninitiated?
- wbl 1mo ago
  The rank of an elliptic curve is expected to be at most one by a combination of a simple heuristic from Cohen and Lenstra and a deep BSD consequence (but I think the case needed is known). However we know of families with rank 15, and sometimes those ranks go up at particular points. The record before this morning was 29 and people suspected that was as high as it got. When we learn how this curve was obtained that might change.
  - pirates 1mo ago
    BSD == Birch and Swinnerton-Dyer I think? I am also uninitiated. I enjoy learning about the millennium prize problems but BSD is one I don’t have a very good understanding of.
    - wbl 1mo ago
      BSD is hard to appreciate because it connects a function that only now is known to have the right properties for the expression on the left hand side to a group we know next to nothing about, plus other mysterious factors each of which could be a few months to really understand. Proving the function exists? That went a bit beyond what was needed for Fermat's Last Theorem. Number theory demands breadth as well as depth. So no surprise it's hard to understand.
  - impendia 1mo ago
    > The record before this morning was 29 and people suspected that was as high as it got. Exactly this. A fundamental question in the subject is, whether elliptic curve ranks are bounded. Contrast with e.g. prime numbers, of which are known to be infinitely many. If you set a new record for the largest known prime, then that's cool but everyone knew there were plenty out there to discover. This paper, by leading experts, https://arxiv.org/abs/1602.01431 https://arxiv.org/abs/1602.01431 made a significant impact in the field, coming up with a heuristic argument for why ranks of elliptic curves should be bounded. The same heuristic suggests, albeit more loosely, that we should perhaps be a little bit surprised to see a curve with rank at least 30. So it's mild evidence that the heuristic itself could be mistaken.
gpvos 1mo ago
BSD = https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture GRH = https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis

104 points by robinhouston 1mo ago

zenburnmyface 1mo ago

Want to understand this more? Read the triplet of books by Ash and Gross, specially Elliptical Tales

nhatcher 1mo ago
Hehe, I really like them. And I always recommend those. It's not always that you have some nice books on deep concepts in mathematics for the "general" public

dwrensha 1mo ago

I'm the maintainer of the linked website. This morning, a mysterious user named "ranksunbounded" submitted the linked curve, which has rank at least thirty. This breaks the previous record of 29 found by Elkies and Klagsbrun in 2024. Nobody knows whether it is possible to achieve arbitrarily high ranks.

impendia 1mo ago
I'm a research mathematician, and to me the biggest surprise is that the user didn't identify themselves or (yet) try to claim credit for their work. Within academia, it is to your definite advantage to have your name associated with breakthroughs, for obvious reasons. Outside academia I don't have firsthand experience, but I'd have to imagine the same holds.
- bdamm 1mo ago
  If this work has an implication for cryptography they might want to be anonymous for any number of reasons.
  - charlieyu1 1mo ago
    Or a random amateur who got lucky with AI. This happened to someone I know and he doesn’t even know how to publish
- shshshhss 1mo ago
  It is also a sign of wisdom and maturity to not care so much, especially outside academia. I know your reaction is the normal one these days, but I’d like to normalize the ones that don’t care about society petting them on the head and saying atta boy. They just deliver the work and retreat into the darkness. I find it highly inspiring. The alternative.. not so much.
  - voakbasda 1mo ago
    Agreed. This is what real research looks like. Not publish or perish. Not going after a bunch of grants. Just doing the work and getting results. Sadly this is this antithesis of modern academia.
- impendia 1mo ago
  Update: the linked page has now been updated to say: > it was Claude, with Levent Alpöge and Ava Howell! Alpöge is the same guy who prompted Claude for a proof of the Jacobian conjecture.
  - fifdspence 1mo ago
    He may have a penchant for unusual announcements. Making an "ranksunbounded" user and posting a 30 today, 31 tomorrow, and 32 the next wouldn't shock me.
- TomatoCo 1mo ago
  You might appreciate the first author on this paper, https://oeis.org/A180632/a180632.pdf https://oeis.org/A180632/a180632.pdf https://en.wikipedia.org/wiki/Superpermutation#Lower_bounds,_or_the_Haruhi_problem https://en.wikipedia.org/wiki/Superpermutation#Lower_bounds,...
  - robinhouston 1mo ago
    Funny coincidence: I (who submitted this story) am the second author.
    - TomatoCo 1mo ago
      I hadn't read carefully enough to recognize that! That's a wacky coincidence that I love. Thank you for your writing.
dwrensha 1mo ago
ranksunbounded has added a comment on the curve: "it was Claude, with Levent Alpöge and Ava Howell!"
- bmenrigh 1mo ago
  I just don't understand how Alpöge can get Claude to construct so many complex mathematical objects. He must already have some deep insight into the structure these things are likely to take. Naive prompting, or naive search would not have found pretty much any of the object he's found in the past month.
  - Resident1324 1mo ago
    He has an AB in Math from Harvard with a senior thesis "The average elliptic curve has few integral points", and a PhD in Math from Princeton with thesis "Points on Curves", so it's fair to say he has some deep insight. (https://alpo.ge/cv.pdf https://alpo.ge/cv.pdf)
    - bmenrigh 1mo ago
      Okay this result may well be in his expertise wheelhouse but what about the recent smooth Carathéodory conjecture disproof? Or the Jacobian, or the Hadamard matrices? It seems to span a pretty wide range. I know others are getting good results too, but his really stand out.
  - ailunicguy 1mo ago
    It wouldn't surprise me if he is given access to a more advanced model or more compute, since he is an Anthropic employee. Add to that the fact that this makes great PR and that he has not shown any of his prompts, it's more than likely in my opinion that this is the case.

great_psy 1mo ago

Can we have a tldr of the implications of this for the uninitiated?

wbl 1mo ago
The rank of an elliptic curve is expected to be at most one by a combination of a simple heuristic from Cohen and Lenstra and a deep BSD consequence (but I think the case needed is known). However we know of families with rank 15, and sometimes those ranks go up at particular points. The record before this morning was 29 and people suspected that was as high as it got. When we learn how this curve was obtained that might change.
- pirates 1mo ago
  BSD == Birch and Swinnerton-Dyer I think? I am also uninitiated. I enjoy learning about the millennium prize problems but BSD is one I don’t have a very good understanding of.
  - wbl 1mo ago
    BSD is hard to appreciate because it connects a function that only now is known to have the right properties for the expression on the left hand side to a group we know next to nothing about, plus other mysterious factors each of which could be a few months to really understand. Proving the function exists? That went a bit beyond what was needed for Fermat's Last Theorem. Number theory demands breadth as well as depth. So no surprise it's hard to understand.
- impendia 1mo ago
  > The record before this morning was 29 and people suspected that was as high as it got. Exactly this. A fundamental question in the subject is, whether elliptic curve ranks are bounded. Contrast with e.g. prime numbers, of which are known to be infinitely many. If you set a new record for the largest known prime, then that's cool but everyone knew there were plenty out there to discover. This paper, by leading experts, https://arxiv.org/abs/1602.01431 https://arxiv.org/abs/1602.01431 made a significant impact in the field, coming up with a heuristic argument for why ranks of elliptic curves should be bounded. The same heuristic suggests, albeit more loosely, that we should perhaps be a little bit surprised to see a curve with rank at least 30. So it's mild evidence that the heuristic itself could be mistaken.

gpvos 1mo ago

BSD = https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture GRH = https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis