*Post by bert*. . . all circles have no more than 12 rational points . . .

Not true. The circle with radius 65 has 20 rational points.

Relative to the origin (0, 0), they are (65, 0) and its three

rotations; (63, 16) and its seven rotations and reflections;

and (33, 56) and its seven rotations and reflections. Now,

really, I thought you knew enough about Pythagorean triangles

to deduce that when the radius is a product of N distinct

primes, all of which are of the form 4m+1, then the circle

has (4 + 2^(N+3)) rational points; that's 12 when N = 1,

20 when N = 2, 36 when N = 3, and so on.

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Hi, Bert, I appreciate your answer, but I think this is much tougher than you think.

Admittedly, both of us have gone through life having been brainwashed about rationals, irrationals, geometry. Brainwashed like that of having to deal with metric, only we grew up on inches, feet, mile.

Admittedly I started off with -- let me explore this -- not with here is a well formed conjecture and so, what I find in a first few posts will not be what I find out later on.

Now Wally in another post mentions the word "lattice" and I need that concept. A concept of finite Rational points of a Grid of Lattice Rationals.

So, I have a Graph paper and focused on the intersection points, and we call each of those points a Rational Number point. The holes in between are irrationals and cannot use them

Now a graph paper is like this

____________

____________

____________

with lines going like this |||||| also

So where the lines

_____

_____

_____ and the lines ||||||||

intersect are points and those points are the only Rationals that exist

Do they make graph paper that has no lines but only points like this

...........

...........

...........

Those points are intersection points of lines that go ___ and lines that go |

So I want graph paper that is only Lattice points.

Now, let us say we drop down to the 10^604 Grid where the spacing between the Rationals is 1^10^-604. We cannot draw anymore points of Rationals in between two points because we are at infinity.

So, Bert, being here at infinity, and not able to ever draw more points of Rationals, we ask the question. Given a Rational number point for center and a radius that is rational number and then draw that circle, how many Rational points will that circle ensnare, not counting the center.

Now Bert thinks it varies according to radius. Bert thinks that a circle can achieve 20 Rational Points.

Now I look up the figure of a 20-gon and ask if there is a irrational number involved. In fact, I know that the 20-gon is the same answer as 10-gon and then 5-gon. I apply the POLYGON PERPENDICULAR method. Where I do a perpendicular from one side to the "opposite side where I have to include at minimum two vertices forming a right triangle. Is the perpendicular a rational number or irrational? In the case of 5-gon, 10-gon, 20-gon the perpendicular is irrational. That tells me automatically that Bert could not possibly have a circle with 20 Rational Lattice points.

Now, I can achieve a circle with 4, 8, 16, 32, 64, etc etc Rational number points, by going to the 10^603 Grid where I call a unit Square having ten by ten smaller squares and what I do then is form a octagon from out of a Square of the 10^603 Grid. Now, to get a circle with 16 Rational points, I have to go to the 10^602 Grid in order to reduce a square in 10^602 to achieve a 16-gon.

Bert, this is tough, this is not easy, for we have to shuck all the brainwash we were taught and try to think straight. It is difficult in shucking miles when we are to use kilometers.

AP