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corrected "contains" to "goes through".
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Gerry Myerson
Gerry Myerson
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How many integer points does my favorite ellipse containgo through?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ x^2+y^2 - \frac{2(d-1)}{d}xy-x-y =0 $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse containsgoes through, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.

How many integer points does my favorite ellipse contain?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ x^2+y^2 - \frac{2(d-1)}{d}xy-x-y =0 $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse contains, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.

How many integer points does my favorite ellipse go through?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ x^2+y^2 - \frac{2(d-1)}{d}xy-x-y =0 $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse goes through, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.

deleted 1 characters in body
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Rainer Sinn
Rainer Sinn
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The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ f = x^2+y^2 - \frac{2(d-1)}{d}xy-x-y $$$$ x^2+y^2 - \frac{2(d-1)}{d}xy-x-y =0 $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse contains, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ f = x^2+y^2 - \frac{2(d-1)}{d}xy-x-y $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse contains, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ x^2+y^2 - \frac{2(d-1)}{d}xy-x-y =0 $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse contains, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.

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Rainer Sinn
Rainer Sinn
  • 174
  • 7

How many integer points does my favorite ellipse contain?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is $$ f = x^2+y^2 - \frac{2(d-1)}{d}xy-x-y $$ By construction, the ellipse is symmetric with respect to the lines $x=y$ and $y=d-x$.

I would like to know exactly how many lattice points this ellipse contains, depending on $d$. Computations suggest that the number is six most of the time and sometimes ten. No other numbers occured. An explanation of that fact (if it's true) is also very welcome.