Balanced prime

<h2 id="examples">Examples</h2>
<p>The first few balanced primes are
</p><p><a href="/facts/5_(number)/gshjM6ID">5</a>, <a href="/facts/53_(number)/nhBUvG9v">53</a>, <a href="/facts/157_(number)/f29abj1G">157</a>, <a href="/facts/173_(number)/XTJenJr8">173</a>, <a href="/facts/211_(number)/gCNxhWYM">211</a>, <a href="/facts/257_(number)/AxyQQQUl">257</a>, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903 (sequence A006562 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).
</p>
<h2 id="infinitude">Infinitude</h2>

Unsolved problem in mathematics:
Are there infinitely many balanced primes?
<a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">(more unsolved problems in mathematics)</a>

<p>It is <a href="/facts/Conjecture/oPWUb7SF">conjectured</a> that there are <a href="/facts/Infinite_set/9rTN0xGH">infinitely many</a> balanced primes.
</p><p>Three consecutive <a href="/facts/Primes_in_arithmetic_progression/goLLlEVE">primes in arithmetic progression</a> is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. As of 2023<a href="https://en.wikipedia.org/w/index.php?title=Balanced_prime&action=edit">[update]</a> the largest known CPAP-3 has 15004 decimal digits and was found by Serge Batalov. It is:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>

p
          
            n
          
        
        =
        2494779036241
        ×
        
          2
          
            49800
          
        
        +
        7
        ,
        
        
          p
          
            n
            −
            1
          
        
        =
        
          p
          
            n
          
        
        −
        6
        ,
        
        
          p
          
            n
            +
            1
          
        
        =
        
          p
          
            n
          
        
        +
        6.
      
    
    {\displaystyle p_{n}=2494779036241\times 2^{49800}+7,\quad p_{n-1}=p_{n}-6,\quad p_{n+1}=p_{n}+6.}

<p>(The value of <i>n</i>, i.e. its position in the sequence of all primes, is not known.)
</p>
<h2 id="generalization">Generalization</h2>
<p>The balanced primes may be generalized to the balanced primes of order <i>n</i>. A balanced prime of order <i>n</i> is a prime number that is equal to the arithmetic mean of the nearest <i>n</i> primes above and below. Algebraically, the 
  
    
      
        k
      
    
    {\displaystyle k}
  
th prime number 
  
    
      
        
          p
          
            k
          
        
      
    
    {\displaystyle p_{k}}
  
 is a balanced prime of order 
  
    
      
        n
      
    
    {\displaystyle n}
  
 if
</p>

p
          
            k
          
        
        =
        
          
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  n
                
              
              (
              
                
                  p
                  
                    k
                    −
                    i
                  
                
                +
                
                  p
                  
                    k
                    +
                    i
                  
                
                )
              
            
            
              2
              n
            
          
        
        .
      
    
    {\displaystyle p_{k}={\sum _{i=1}^{n}({p_{k-i}+p_{k+i})} \over 2n}.}

<p>Thus, an ordinary balanced prime is a balanced prime of order 1. The sequences of balanced primes of orders 2, 3, and 4 are A082077, A082078, and A082079 in the OEIS respectively.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Strong_prime/9XeCXPX5">Strong prime</a>, a prime that is greater than the arithmetic mean of its two neighboring primes</li>
<li><a href="/facts/Interprime/Pl1y7Lhs">Interprime</a>, a composite number balanced between two prime neighbours</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>The Largest Known CPAP's. Retrieved on 2023-01-06. <a href="http://primerecords.dk/cpap.htm#k3" target="_blank">http://primerecords.dk/cpap.htm#k3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Balanced prime open-in-new

Balanced prime