Bailey–Borwein–Plouffe formula

The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for <a href="/facts/Pi/vC0UBj1q">π</a>. It was discovered in 1995 by <a href="/facts/Simon_Plouffe/JeISXKeT">Simon Plouffe</a> and is named after the authors of the article in which it was published, <a href="/facts/David_H._Bailey_(mathematician)/0rhMd4Jz">David H. Bailey</a>, <a href="/facts/Peter_Borwein/dJtVFCLg">Peter Borwein</a>, and Plouffe. The formula is:

π
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          [
          
            
              
                1
                
                  16
                  
                    k
                  
                
              
            
            
              (
              
                
                  
                    4
                    
                      8
                      k
                      +
                      1
                    
                  
                
                −
                
                  
                    2
                    
                      8
                      k
                      +
                      4
                    
                  
                
                −
                
                  
                    1
                    
                      8
                      k
                      +
                      5
                    
                  
                
                −
                
                  
                    1
                    
                      8
                      k
                      +
                      6
                    
                  
                
              
              )
            
          
          ]
        
      
    
    {\displaystyle \pi =\sum _{k=0}^{\infty }\left[{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)\right]}

The BBP formula gives rise to a <a href="/facts/Spigot_algorithm/erV8opXS">spigot algorithm</a> for computing the nth <a href="/facts/Hexadecimal/02ohQW22">base-16</a> (hexadecimal) digit of π (and therefore also the 4nth <a href="/facts/Binary_digit/S972NSaD">binary digit</a> of π) without computing the preceding digits. This does not compute the nth decimal digit of π (i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting the nth digit of π in decimal. BBP and BBP-inspired algorithms have been used in projects such as <a href="/facts/PiHex/HkR93cEG">PiHex</a> for calculating many digits of π using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the nth digit of π is just as hard as computing the first n digits.
Since its discovery, formulas of the general form:

α
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          [
          
            
              
                1
                
                  b
                  
                    k
                  
                
              
            
            
              
                
                  p
                  (
                  k
                  )
                
                
                  q
                  (
                  k
                  )
                
              
            
          
          ]
        
      
    
    {\displaystyle \alpha =\sum _{k=0}^{\infty }\left[{\frac {1}{b^{k}}}{\frac {p(k)}{q(k)}}\right]}

have been discovered for many other <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a> 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
, where 
 
 
 
 p
 (
 k
 )
 
 
 {\displaystyle p(k)}
 
 and 
 
 
 
 q
 (
 k
 )
 
 
 {\displaystyle q(k)}
 
 are <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> with integer coefficients and 
 
 
 
 b
 ≥
 2
 
 
 {\displaystyle b\geq 2}
 
 is an integer <a href="/facts/Radix/J5RGWRJF">base</a>.
Formulas of this form are known as BBP-type formulas. Given a number 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
, there is no known systematic algorithm for finding appropriate 
 
 
 
 p
 (
 k
 )
 
 
 {\displaystyle p(k)}
 
, 
 
 
 
 q
 (
 k
 )
 
 
 {\displaystyle q(k)}
 
, and 
 
 
 
 b
 
 
 {\displaystyle b}
 
; such formulas are discovered <a href="/facts/Experimental_mathematics/NfuxNMIz">experimentally</a>.

Bailey–Borwein–Plouffe formula open-in-new

Bailey–Borwein–Plouffe formula