Hearted Youtube comments on Mathologer (@Mathologer) channel.
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You can also use the golden ratio to drastically improve the prime number theorem.
Number of primes less than (n/phi)^2=((n-n/phi)x(n/Ln(n-1))/2.
For example: n=641,894
Pi(157,380,656,251)
=6,361,970,514
n/Ln(n-1)=6,350,670,612
Using the golden ratio=6,360,264,553
Not sure if this shows some connection between phi, e and primes or just a neat trick that works really well.
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2:34, check of the calculations up to that point:
For the equilateral triangle, the height is the length of one of its legs times the cosine of π/6, so the height equals 2*sqrt(3) times sqrt(3)/2 .
Thus, the area is half its base, i. e. sqrt(3), times 2 times sqrt(3) times sqrt(3) times 1/2 which amounts to 3*sqrt(3).
For the isosceles triangle with legs of the length 2+φ with a base of 2*φ, Pythagoras's theorem gives sqrt( (2+φ)^2 - φ^2 ) = sqrt(4 + 4φ + φ^2 - φ^2) = sqrt( 4(1+φ) ) .
The golden ratio is a solution to the the equation x^2 = x+1 , so sqrt( 4(1+φ) ) = sqrt( 4φ^2 ) = 2φ .
(Since we're talking about lengths, we can ignore negative results.)
Finally, the area of this isosceles triangle is then half its base times its height: φ times 2φ = 2φ^2 .
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