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 Register FAQ Search Today's Posts Mark Forums Read  2016-09-07, 21:11 #23 CRGreathouse   Aug 2006 3·1,993 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 22×32×11 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

10111010110112 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.   Thread Tools Show Printable Version Email this Page

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