+4609 A circle of radius 5 with its center at (0,0) is drawn on a Cartesian coordinate system. How many lattice points (points with integer coordinates) lie within or on this circle?
+26367 A circle of radius 5 with its center at (0,0) is drawn on a Cartesian coordinate system.
How many lattice points (points with integer coordinates) lie within or on this circle?
Let K the circle with center (0,0)
Let r the radius of the circle.
Let g(r) are the lattice points lie within or on this circle. \([~g(r) :=|{(x,y)\in \mathbb{Z^2}|x^2+y^2\le r^2}|~]\)
We have four subsets A, B, C, and D with g(r) = A+B+C+D
Subset D:
For D there is no Formula in close form, but though we have:
\(\frac{D}{8} = [\sqrt{r^2 - a^2}] +[\sqrt{r^2 - (a+1)^2}] + \cdots + [\sqrt{r^2 - (r-1)^2}] \) with \(a =\left[ \dfrac{r}{\sqrt2} \right] + 1\) and
\([x] =\) greatest integer number
g(5) = ?
\(\begin{array}{|rcll|} \hline A &=& 1 \\ B &=& 4\cdot 5 \\ &=& 20 \\ C &=& 9\cdot 4 \quad &| \quad \left[ \dfrac{r}{\sqrt2} \right] = \left[ \dfrac{5}{\sqrt2} \right] = 3 \\ &=& 36 \\ D &=& 8\cdot ( [\sqrt{5^2 - 4^2}] ) \quad &| \quad a =\left[ \dfrac{5}{\sqrt2} \right] + 1 = 3 +1 = 4 \qquad r-1 = 5-1 = 4 \\ &=& 8\cdot ( [\sqrt{25 - 16}] ) \\ &=& 8\cdot ( [\sqrt{9}] ) \\ &=& 8\cdot [3] \\ &=& 8\cdot 3 \\ &=&24\\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline g(5) &=& A+B+C+D \\ &=& 1 + 20 + 36 + 24 \\ &=& 81 \\ \hline \end{array}\)
There are 81 lattice points.
Source: http://www.matheprisma.uni-wuppertal.de/Module/PIXXL/Worksheet/ws.pdf
+128474 I believe this related to something known as Gauss's Circle Problem....however...I think Gauss was only concerned with the lattice points inside the circle...
The problem actually boils down to finding how many integer pairs (m,n) exist such that:
m^2 + n^2 ≤ r^2
Anyway, using "brute force," there are 35 lattice points above the x axis.....so by symmetry, there are also 35 below
And 11 lattice points lie on the x axis....so we have.....
[ 35 + 35 + 11 ] = 81 {If I counted correctly....!!! }
Note that, if we assume that each square unit of the circle contains one lattice point....then we should have about 25*pi ≈ 79 points....so....81 is pretty close to this.....
P.S. - Maybe heureka or Alan can enlighten us with a presentation of the exact formula for this....I don't know enough higher math to carry it out ....!!!!
+26367 A circle of radius 5 with its center at (0,0) is drawn on a Cartesian coordinate system.
How many lattice points (points with integer coordinates) lie within or on this circle?
Let K the circle with center (0,0)
Let r the radius of the circle.
Let g(r) are the lattice points lie within or on this circle. \([~g(r) :=|{(x,y)\in \mathbb{Z^2}|x^2+y^2\le r^2}|~]\)
We have four subsets A, B, C, and D with g(r) = A+B+C+D
Subset D:
For D there is no Formula in close form, but though we have:
\(\frac{D}{8} = [\sqrt{r^2 - a^2}] +[\sqrt{r^2 - (a+1)^2}] + \cdots + [\sqrt{r^2 - (r-1)^2}] \) with \(a =\left[ \dfrac{r}{\sqrt2} \right] + 1\) and
\([x] =\) greatest integer number
g(5) = ?
\(\begin{array}{|rcll|} \hline A &=& 1 \\ B &=& 4\cdot 5 \\ &=& 20 \\ C &=& 9\cdot 4 \quad &| \quad \left[ \dfrac{r}{\sqrt2} \right] = \left[ \dfrac{5}{\sqrt2} \right] = 3 \\ &=& 36 \\ D &=& 8\cdot ( [\sqrt{5^2 - 4^2}] ) \quad &| \quad a =\left[ \dfrac{5}{\sqrt2} \right] + 1 = 3 +1 = 4 \qquad r-1 = 5-1 = 4 \\ &=& 8\cdot ( [\sqrt{25 - 16}] ) \\ &=& 8\cdot ( [\sqrt{9}] ) \\ &=& 8\cdot [3] \\ &=& 8\cdot 3 \\ &=&24\\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline g(5) &=& A+B+C+D \\ &=& 1 + 20 + 36 + 24 \\ &=& 81 \\ \hline \end{array}\)
There are 81 lattice points.
Source: http://www.matheprisma.uni-wuppertal.de/Module/PIXXL/Worksheet/ws.pdf
+26367 Calculation of the Number of Lattice Points in the Circle
Let \(r\) be the Radius of the Circle
Let \(x = r^2\)
Let \( A_2(x)\) be the number of lattice points in circle
Let \([\ldots] \) be the largest integer
Noted by Gauss:
\(\begin{array}{|rcll|} \hline A_2(x) &=& 1 + 4\cdot \left[~ \sqrt x ~\right] + 4 \cdot \left[~ \sqrt{\frac{x}{2}} ~\right]^2 + \sum \limits_{y_1=\left[~ \sqrt{\frac{x}{2}} ~\right]+1}^{\left[~ \sqrt x ~\right]} \left[~ \sqrt{x-y_1^2} ~\right]\\ \hline \end{array}\)
Computed Results
\(\begin{array}{r|r} \hline r = \sqrt{x} & A_2(x) \\ \hline 1&5\\ 2&13\\ 3&29\\ 4&49\\ 5&81\\ 6&113\\ 7&149\\ 8&197\\ 9&253\\ 10&317\\ 11&377\\ 12&441\\ 13&529\\ 14&613\\ 15&709\\ 16&797\\ 17&901\\ 18&1009\\ 19&1129\\ 20&1257\\ 21&1373\\ 22&1517\\ 23&1653\\ 24&1793\\ 25&1961\\ 26&2121\\ 27&2289\\ 28&2453\\ 29&2629\\ 30&2821\\ 31&3001\\ 32&3209\\ 33&3409\\ 34&3625\\ 35&3853\\ 36&4053\\ 37&4293\\ 38&4513\\ 39&4777\\ 40&5025\\ 41&5261\\ 42&5525\\ 43&5789\\ 44&6077\\ 45&6361\\ 46&6625\\ 47&6921\\ 48&7213\\ 49&7525\\ \hline \end{array}\)
Continued
\(\begin{array}{r|r} \hline r = \sqrt{x} & A_2(x) \\ \hline 50&7845\\ 51&8173\\ 52&8497\\ 53&8809\\ 54&9145\\ 55&9477\\ 56&9845\\ 57&10189\\ 58&10557\\ 59&10913\\ 60&11289\\ 61&11681\\ 62&12061\\ 63&12453\\ 64&12853\\ 65&13273\\ 66&13673\\ 67&14073\\ 68&14505\\ 69&14949\\ 70&15373\\ 71&15813\\ 72&16241\\ 73&16729\\ 74&17193\\ 75&17665\\ 76&18125\\ 77&18605\\ 78&19109\\ 79&19577\\ 80&20081\\ 81&20593\\ 82&21101\\ 83&21629\\ 84&22133\\ 85&22701\\ 86&23217\\ 87&23769\\ 88&24313\\ 89&24845\\ 90&25445\\ 91&25997\\ 92&26565\\ 93&27145\\ 94&27729\\ 95&28345\\ 96&28917\\ 97&29525\\ 98&30149\\ 99&30757\\ 100&31417\\ \hline \end{array}\)
+26367 The equation noted by Gauss should be of course:
\(\begin{array}{|rcll|} \hline A_2(x) &=& 1 + 4\cdot \left[~ \sqrt x ~\right] + 4 \cdot \left[~ \sqrt{\frac{x}{2}} ~\right]^2 + \ 8 \sum \limits_{y_1=\left[~ \sqrt{\frac{x}{2}} ~\right]+1}^{\left[~ \sqrt x ~\right]} \left[~ \sqrt{x-y_1^2} ~\right]\\ \hline \end{array}\)
