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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?

tertre  Mar 14, 2017

Best Answer 

 #3
avatar+18777 
+5

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)

Letthe 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

  • A = {(0,0)} = 1   
  • B = Cut K with axis without (0,0) \(= 2\cdot r + 2\cdot r = 4\cdot r \) 
  • \(C_1, C_2, C_3, C_4 =\) squares parallel to the axis with one corner in (0,0) and edge length \(\frac{r}{\sqrt2}\)   
    \(= 4\cdot \left( \left[ \dfrac{r}{\sqrt2} \right] \right)^2\)   with \([x] = \) greatest integer number   
  •  D = remain

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

 

laugh

heureka  Mar 15, 2017
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5+0 Answers

 #1
avatar+79881 
+5

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 ....!!!!

 

 

cool cool cool

CPhill  Mar 14, 2017
 #2
avatar+1227 
0

Thank you so much!

tertre  Mar 14, 2017
 #3
avatar+18777 
+5
Best Answer

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)

Letthe 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

  • A = {(0,0)} = 1   
  • B = Cut K with axis without (0,0) \(= 2\cdot r + 2\cdot r = 4\cdot r \) 
  • \(C_1, C_2, C_3, C_4 =\) squares parallel to the axis with one corner in (0,0) and edge length \(\frac{r}{\sqrt2}\)   
    \(= 4\cdot \left( \left[ \dfrac{r}{\sqrt2} \right] \right)^2\)   with \([x] = \) greatest integer number   
  •  D = remain

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

 

laugh

heureka  Mar 15, 2017
 #4
avatar+18777 
0

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

\(\begin{array}{r|r} \hline r = \sqrt{x} & A_2(x) \\ \hline 50&7845\\ 51&8173\\ 52&8497\\ 53&8809\\ 54&9145\\ 55&9477\\ 56&9845\\ 57&10189\\ 58&10557\\ 59&10913\\ 60&11289\\ 61&11681\\ 62&12061\\ 63&12453\\ 64&12853\\ 65&13273\\ 66&13673\\ 67&14073\\ 68&14505\\ 69&14949\\ 70&15373\\ 71&15813\\ 72&16241\\ 73&16729\\ 74&17193\\ 75&17665\\ 76&18125\\ 77&18605\\ 78&19109\\ 79&19577\\ 80&20081\\ 81&20593\\ 82&21101\\ 83&21629\\ 84&22133\\ 85&22701\\ 86&23217\\ 87&23769\\ 88&24313\\ 89&24845\\ 90&25445\\ 91&25997\\ 92&26565\\ 93&27145\\ 94&27729\\ 95&28345\\ 96&28917\\ 97&29525\\ 98&30149\\ 99&30757\\ 100&31417\\ \hline \end{array}\)

 

Source: http://http://www.ams.org/journals/mcom/1962-16-079/S0025-5718-1962-0155788-9/S0025-5718-1962-0155788-9.pdf

 

laugh

heureka  Mar 16, 2017
 #5
avatar+18777 
+5

The equation noted by Gauss should be of course:

 

\(\begin{array}{|rcll|} \hline A_2(x) &=& 1 + 4\cdot \left[~ \sqrt x ~\right] + 4 \cdot \left[~ \sqrt{\frac{x}{2}} ~\right]^2 + \ 8 \sum \limits_{y_1=\left[~ \sqrt{\frac{x}{2}} ~\right]+1}^{\left[~ \sqrt x ~\right]} \left[~ \sqrt{x-y_1^2} ~\right]\\ \hline \end{array}\)

 

laugh

heureka  Mar 16, 2017

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