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.
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.
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.
such a good lecture to let non-expert to get some sense of number theory
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.
Table at 26:00 shows the solutions for p=41 are13 and 29. It should have written 13 and 28.
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.
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
Good video, small mistake in the table at 24'30"; 29 squared is 21 (mod 41)
At 30:35, you used the reciprocity law wrongly, it should be "x^2 = 7 modulo 12".
@25:45 I see an alternative pattern through primes which is amazing :)
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.
great job by a great professor! this really cleared up a bunch of concepts ... love that he included so many examples!
in fact it is possible to find an algorithm to solve quadratic equations the congruent number problem is a special case.
Thanks !
Love from India
waw ! interesting stuff.
it's hard to explain the stuff you don't know about...