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Take a spiral of integers:

5 4 3
6 1 2
7 8 9

Hilite the prime numbers, putting a bullet at the origin for reference:

X - X
- • X
X - -

And you have a prime spiral. Here's a big one:

              X   X X   X      
             X   X       X   X 
  X   X           X X     X    
 X X     X X     X     X       
          X           X   X    
     X   X       X     X       
X   X         X   X X   X X X  
       X     X         X X   X 
  X     X   X X               X
               X X     X   X   
X   X   X X   X       X   X X  
               X X   X         
      X   X X     X X X     X X
 X X X X X X X   X       X     
              X X X           X
         X   XXX X X X   X X 
  X       X X X                
           X   X               
X   X X   X X   X   X X   X   X
   X   X   X     X     X X   X 
            X           X      
       X X     X   X   X       
    X   X           X       X  
     X     X   X X             
          X X   X     X   X    
 X X X         X X     X     X 
  X   X           X X          
 X X     X     X   X X         
X       X         X       X    
         X X   X X         X   
      X   X     X     X        

On a large scale, patterns emerge. There are broken lines, diagonal lines, interesting shapes. These can give formulas for predicting primes. Take, for example, the broken line leading left from 17. It can be approximated by the formula f(n) =16 n2 + 2 n - 1, which yields prime numbers for small values of n:

n       1    2    3    4    5    6    7    8    9   10
f(n)   17   67  149  263  409  587  797 1039 1313 1619
prime?  y    y    y    y    y    y    y    y    n    y

It breaks down at n = 9. All such formulas eventually break down.

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