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Mar 16, 2017
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Mar 16, 2017
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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

Mar 16, 2017

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