Murmuration Math

Patterns formed by thousands of elliptic curves bear a striking similarity to murmurations of starlings.

An elliptic curve has an associated sequence of numbers, called ap, which relates to the number of solutions there are to the curve in the finite field defined by the prime p. A smaller ap means more solutions; a bigger ap means fewer solutions. Though the rank is hard to calculate, the sequence ap is a lot easier.

On the basis of numerous calculations done on one of the very first computers, Birch and Swinnerton-Dyer conjectured a relationship between an elliptic curve’s rank and the sequence ap. Anyone who can prove they were right stands to win a million dollars and mathematical immortality.

Elliptic Curve ‘Murmurations’ Found With AI Take Flight

Murmurations have been mentioned here several times. I don’t know if this has anything to do with birds but the similarity is intriguing. I read the article. If anyone can explain what they are talking about I am up for it.

1 comment

  1. This is all very well and good and so far over my head I wouldn’t even know where to begin. But I wonder, they are dealing with pure math, what happens when natural constants enter into the equation. Things like the shape of a starlings wing and how that cuts the air. Is the curve or murmurations of the flock. How much energy is used or conserved doing these patterns. Yes its all about confusing a hawk or falcons eye but chances are someone’s going to get munched. So perhaps more AI needed. And this is to much thinking for Friday and I wonder when some one will apply this to drone swarming if they already haven’t.

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