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 2016-09-07, 21:11 #23 CRGreathouse     Aug 2006 31×191 Posts Here's a heuristic suggesting that my constant is right. Start by looking at the circle with radius sqrt(x). All lattice points in the circle are (u, v) with u^2 + v^2 < x. The circle has area $$\pi x$$. Now look only at the first quadrant where u, v > 0, a quarter-circle of area $$\pi x/4$$. But we need to split it in half to get only the values with 0 < u < v which has area $$\pi x/8.$$ Notice that the number of lattice points inside this area is equal to its area plus an error term equal to its outside (perimeter/circumference/etc.) for a total of $$\pi x/8+O(\sqrt x)$$. So far this is a rigorous count of the number of points, but the next step is just a heuristic. The density of squarefree numbers is $$6/\pi^2,$$ so the expected number of 0 < u < v with gcd(u, v) = 1 and u^2 + v^2 < x is about $$\pi x/8\cdot6/\pi^2 = \frac{3}{4\pi}x$$ as desired. The last step can probably be made rigorous -- the area is pretty regular and the error bounds on squarefree density are small.
 2016-09-08, 10:24 #24 bhelmes     Mar 2016 22×67 Posts A peaceful day for you, thank you for your efforts and explications. i know where the problem comes from: u should be odd and v even, or inverse ! so the tripple (u,v)=(1,3) is not mentionned in my list. i learned that for primitiv pyth. tripples that this fact is necessary, but i did not mentionned it, it was my fault. I will research if the amount of pyth. triples or the amount of different roots of 1 deliver a better approximation of pi. Nice greetings from the primes Bernhard
2016-09-08, 14:03   #25
CRGreathouse

Aug 2006

31·191 Posts

Quote:
 Originally Posted by bhelmes u should be odd and v even, or inverse ! so the tripple (u,v)=(1,3) is not mentionned in my list. i learned that for primitiv pyth. tripples that this fact is necessary, but i did not mentionned it, it was my fault.
If you're going to arbitrarily reject some values then you can adapt my method and see if it comes out as desired.