Primes and Equations | Richard Taylor

Richard Taylor, Professor, School of Mathematics, Institute for Advanced Study
www.ias.edu/peo...
One of the oldest subjects in mathematics is the study of Diophantine equations, i.e., the study of whole number (or fractional) solutions to polynomial equations. It remains one of the most active areas of mathematics today. Perhaps the most basic tool is the simple idea of "congruences," particularly congruences modulo a prime number. In this talk, Richard Taylor, Professor in the School of Mathematics, introduces prime numbers and congruences and illustrates their connection to Diophantine equations. He also describes recent progress in this area, an application, and reciprocity laws, which lie at the heart of much recent progress on Diophantine equations, including Wiles's proof of Fermat's last theorem.
More videos at video.ias.edu

[@fengguan7444]

such a good lecture to let non-expert to get some sense of number theory

[@aileenwu1362]

great job by a great professor! this really cleared up a bunch of concepts ... love that he included so many examples!

[@paul1964uk]

Interesting topic. Didn't think all the connections were sufficiently clearly made however. Still kudos to the Prof for giving a talk as accessible as this was.

[@geraldillo]

Good video, small mistake in the table at 24'30"; 29 squared is 21 (mod 41)

[@stereosphere]

14:07 Another Diophantine Problem Dr Taylor is scaling the radius by 999999/1000000 which has the effect or scaling x and y by the square root of 999999/1000000, an irrational number. If he scaled by the square root of 9999999/1000000, the rational numbers would reappear. The slide he skips over at 15:09 shows how a circle can be formed from rational coordinates, as long as the radius is a perfect square. s = 0.999999 q = sqrt(0.999999) x^2 + y^2 = 1 s*x^2 + s*y^2 = s (q*x)^2 + (q*y)^2 = s If x is rational, q*x is irrational. Same for y.

[@marcderiveau9307]

Table at 26:00 shows the solutions for p=41 are13 and 29. It should have written 13 and 28.

[@chrisholding2382]

@25:45 I see an alternative pattern through primes which is amazing :)

[@gurmeet0108]

At 30:35, you used the reciprocity law wrongly, it should be "x^2 = 7 modulo 12".

[@naimulhaq9626]

Extremely illustrative of the topic, thank you very much. Gauss' reciprocity law implies not only Wiles but a host of properties of numbers !!! Amazing, how numbers can be mesmerizing !!! There are less and less primes as n increases and at infinity there are none.

[@evid-rz3nu]

25:40----i see a pattern that is 11=4+7. 11/2>4>under root11 And11/2 under root 11 19=9+10. 19/2>9>under root19 and 19under root 19. And so like this

[@123must]

Thanks !

[@evid-rz3nu]

Love from India

[@robkim55]

in fact it is possible to find an algorithm to solve quadratic equations the congruent number problem is a special case.

[@erikcools891]

waw ! interesting stuff.

[@carolinemurgue8170]

[@AdrianReef]

it's hard to explain the stuff you don't know about...