Find coordinates of a prime number in a Prime Spiral

Introduction
The Ulam spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanislaw Ulam. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.You can read more about it here.

But we are going to compute on the alternate version of this spiral where the prime numbers are lined up in the spiral rather than natural numbers as in the original Ulam’s spiral.

The prime numbers are written in a spiral form starting from the origin (0, 0) and moving as shown in the diagram above. The numbers shown in the right column and the bottom row are the column numbers and row numbers respectively ( i.e. y and x coordinates)

The objective is to find the position (x and y coordinates) of a given prime number.

Examples:

Input : 5
Output : 1 1 
As per the diagram above 5 corresponds to 
the column 1 and row 1.

Input : 11
Output : -1 1 
Similarly, 11 will correspond to column -1
and row 1. 



Approach: The general solution to the problem would be to create an algorithm to predict the position of the moving spiral point at every step and reference each step in the spiral to a prime number (e.g. step 0 –> 2; step 1 –> 3 ; step 2 –> 5 and so on). Using this prime number we can trace back to the co-ordinates which is our solution (2 –> [0, 0], 3 –> [1, 0], 5 –> [1, 1]).

We begin by keeping count of x and y, for step k = 1 : x = y = 0. k = 2 leads to increase in x by 1 (x = 1) next comes k = 3 providing y = 1. Now as the spiral rotates we are to decrease the values now by 2. The consecutive movements are shown below:

From the above computations, we observe that the operations are same for a batch of steps, e.g. at first K = 2 & 3 follow one where x is increased by 1 and then y by 1. Next in K= 4, 5, 6, 7 for 4 and 5, x is decreased and y is decreased in K = 6 and 7. Again From K = 8, 9 and 10, x is increased in repeated steps and 11, 12, 13 consists of decreasing y.

Each step can be classified into a batch and the size of batch increases by 1 after two runs. After a run, the operation is switched from y to x or vice versa and after every two batch runs the addition operation is switched to subtraction and vice versa. The following table explains it.

Lastly, we also need to create a prime number generator that will keep on feeding the above algorithm with a prime number that will be referenced to a corresponding step and it will give us the required coordinates.

Here is the implementation of above idea :
Input Limits : 2 < N < 1000000
Assumption : Prime numbers are the only inputs to the program.

C++

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// C++ Program to find coordinates of a
// prime number in a Prime Spiral
#include <bits/stdc++.h>
  
using namespace std;
  
// The main algorithm that keeps track of
// coordinates
void spiralSplicer(int input)
{
    // Batch size tracker
    int step_count = 1;
  
    // Batch size limiter
    int step_limit = 2;
  
    // switches between -1 and +1
    int adder = 1;
  
    // Co-ordinates of step K
    int x = 0, y = 0;
  
    for (int n = 2; n != input + 1; n++, 
                       step_count++) { 
  
        // Keeps track of steps
        // Checks on the current batch
        if (step_count <= .5 * step_limit)
            x += adder; // Switch to operating on x
  
        else if (step_count <= step_limit)
            y += adder; // Switch to operating on x
          
        if (step_count == step_limit) {
   
            // Changes adder to -1 and vice versa
            adder *= -1; 
  
           // Keeps on updating 'step_limit'
           step_limit += 2; 
             
           // Resets count
           step_count = 0; 
        }
    }
    cout << x << " " << y;
}
  
int primeIndex(int input)
{
    int j, cnt, prime_c = 0;
    for (int i = 2; i <= 1000000; i++) {
        cnt = 0;
        for (j = 2; j <= i; j++) {
            if (i % j == 0)
                cnt++;
        }
  
        if (cnt == 1) {
            prime_c++;
  
            if (input == i) {
  
           /* Replaces the prime number with 
              Step K which will be fed into
              the main algorithm*/
                input = prime_c; 
                break;
            }
        }
    }
    return input;
}
  
// driver code
int main()
{
    int input = 113;
  
    // Prime Index Finder Output ported
    // to final algorithm
    spiralSplicer(primeIndex(input));
}

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Java

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// java Program to find coordinates of a
// prime number in a Prime Spiral
public class GFG {
      
    // The main algorithm that keeps track of
    // coordinates
    static void spiralSplicer(int input)
    {
          
        // Batch size tracker
        int step_count = 1;
      
        // Batch size limiter
        int step_limit = 2;
      
        // switches between -1 and +1
        int adder = 1;
      
        // Co-ordinates of step K
        int x = 0, y = 0;
      
        for (int n = 2; n != input + 1
                    n++, step_count++)
        
      
            // Keeps track of steps
            // Checks on the current batch
            if (step_count <= .5 * step_limit)
              
                // Switch to operating on x
                x += adder;
      
            else if (step_count <= step_limit)
              
                // Switch to operating on x
                y += adder;
              
            if (step_count == step_limit)
            {
      
                // Changes adder to -1 and
                // vice versa
                adder *= -1
      
                // Keeps on updating 'step_limit'
                step_limit += 2
                      
                // Resets count
                step_count = 0
            }
        }
          
        System.out.print( x + " " + y);
    }
      
    static int primeIndex(int input)
    {
        int j, cnt, prime_c = 0;
        for (int i = 2; i <= 1000000; i++) 
        {
            cnt = 0;
            for (j = 2; j <= i; j++)
            {
                if (i % j == 0)
                    cnt++;
            }
      
            if (cnt == 1
            {
                prime_c++;
      
                if (input == i) 
                {
      
                    /* Replaces the prime 
                    number with Step K which 
                    will be fed into
                    the main algorithm*/
                    input = prime_c; 
                    break;
                }
            }
        }
          
        return input;
    }     
      
    // Driver code
    public static void main(String args[]) {
  
        int input = 113;
  
        // Prime Index Finder Output ported
        // to final algorithm
        spiralSplicer(primeIndex(input));
  
    }
}
  
// This code is contributed by Sam007.

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Python

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# Python Program to find coordinates of a
# prime number in a Prime Spiral
  
# The main algorithm that keeps track of
# coordinates
def spiralSplicer(inp):
     
    # Batch size tracker
    step_count = 1
   
    # Batch size limiter
    step_limit = 2
   
    # switches between -1 and +1
    adder = 1
   
    # Co-ordinates of step K
    x, y = 0, 0
   
    for n in range(2, inp + 1):
     
        # Keeps track of steps
        # Checks on the current batch
        if (step_count <= .5 * step_limit):
            x += adder # Switch to operating on x
   
        elif (step_count <= step_limit):
            y += adder # Switch to operating on x
           
        if (step_count == step_limit):
            # Changes adder to -1 and vice versa
            adder *= -1
   
            # Keeps on updating 'step_limit'
            step_limit += 2
              
            # Resets count
            step_count = 0
        step_count += 1
    print x, y
   
def primeIndex(inp):
    cnt, prime_c = 0, 0
    for i in range(2, 1000000 + 1):
        cnt = 0
        for j in range(2, i + 1):
            if (i % j == 0):
                cnt += 1
          
        if (cnt == 1):
            prime_c += 1
   
            if (inp == i):
   
                """ Replaces the prime number with 
                    Step K which will be fed into
                    the main algorithm """
                inp = prime_c
                break
    return inp
   
# driver code
inp = 113
  
# Prime Index Finder Output ported
# to final algorithm
temp = primeIndex(inp)
  
spiralSplicer(temp)
  
#This code is contributed by Sachin Bisht

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C#

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// C# Program to find coordinates of a
// prime number in a Prime Spiral
using System;
  
class GFG {
      
    // The main algorithm that keeps track of
    // coordinates
    static void spiralSplicer(int input)
    {
          
        // Batch size tracker
        int step_count = 1;
      
        // Batch size limiter
        int step_limit = 2;
      
        // switches between -1 and +1
        int adder = 1;
      
        // Co-ordinates of step K
        int x = 0, y = 0;
      
        for (int n = 2; n != input + 1; 
                     n++, step_count++)
        
      
            // Keeps track of steps
            // Checks on the current batch
            if (step_count <= .5 * step_limit)
              
                // Switch to operating on x
                x += adder;
      
            else if (step_count <= step_limit)
              
                // Switch to operating on x
                y += adder;
              
            if (step_count == step_limit)
            {
      
                // Changes adder to -1 and
                // vice versa
                adder *= -1; 
      
                // Keeps on updating 'step_limit'
                step_limit += 2; 
                      
                // Resets count
                step_count = 0; 
            }
        }
          
        Console.Write( x + " " + y);
    }
      
    static int primeIndex(int input)
    {
        int j, cnt, prime_c = 0;
        for (int i = 2; i <= 1000000; i++) 
        {
            cnt = 0;
            for (j = 2; j <= i; j++)
            {
                if (i % j == 0)
                    cnt++;
            }
      
            if (cnt == 1) 
            {
                prime_c++;
      
                if (input == i) 
                {
      
                    /* Replaces the prime 
                    number with Step K which 
                    will be fed into
                    the main algorithm*/
                    input = prime_c; 
                    break;
                }
            }
        }
          
        return input;
    }     
      
    // Driver code
    public static void Main ()
    {
        int input = 113;
  
        // Prime Index Finder Output ported
        // to final algorithm
        spiralSplicer(primeIndex(input));
    }        
}
  
// This code is contributed by Sam007.

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PHP

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<?php
// PHP Program to find coordinates of a
// prime number in a Prime Spiral
  
// The main algorithm that
// keeps track of
// coordinates
function spiralSplicer($input)
{
      
    // Batch size tracker
    $step_count = 1;
  
    // Batch size limiter
    $step_limit = 2;
  
    // switches between -1 and +1
    $adder = 1;
  
    // Co-ordinates of step K
    $x = 0; $y = 0;
  
    for ($n = 2; $n != $input + 1; $n++, 
                           $step_count++) 
    
  
        // Keeps track of steps
        // Checks on the current batch
        if ($step_count <= .5 * $step_limit)
          
            // Switch to operating on x
            $x += $adder
  
        else if ($step_count <= $step_limit)
          
            // Switch to operating on x
            $y += $adder
          
        if ($step_count == $step_limit
        {
  
            // Changes adder to -1 
            // and vice versa
            $adder *= -1; 
  
        // Keeps on updating
        // 'step_limit'
        $step_limit += 2; 
              
        // Resets count
        $step_count = 0; 
        }
    }
    echo $x, " ", $y;
}
  
function primeIndex($input)
{
    $j
    $cnt
    $prime_c = 0;
    for ($i = 2; $i <= 1000000; $i++) 
    {
        $cnt = 0;
        for ($j = 2; $j <= $i; $j++)
        {
            if ($i % $j == 0)
                $cnt++;
        }
  
        if ($cnt == 1)
        {
            $prime_c++;
  
            if ($input == $i
            {
  
                // Replaces the prime 
                // number with Step K
                // which will be fed into
                // the main algorithm
                    $input = $prime_c
                    break;
            }
        }
    }
    return $input;
}
  
    // Driver Code
    $input = 113;
  
    // Prime Index Finder Output ported
    // to final algorithm
    spiralSplicer(primeIndex($input));
      
// This Code is contributed by ajit
?>

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Output :

3 2

This article is contributed by Pritom Gogoi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Improved By : Sam007, jit_t