Trong ly thuyet so, so nguyen to Wolstenholme la loai so nguyen to đac biet thoa man dang manh hon cua đinh ly Wolstenholme. Đinh ly Wolstenholme la quan he đong du đuoc thoa man boi moi so nguyen to lon hon ba. Cac so nguyen to Wolstenholme đuoc đat ten theo nha toan hoc Joseph Wolstenholme, nguoi lan đau phat bieu đinh ly tren trong the ky 19.
Cac so nguyen to nay đuoc chu y toi boi quan he cua chung voi đinh ly lon Fermat. Ngoai ra cac so nguyen to Wolstenholme cung co quan he voi mot so lop nguyen to khac.
Chi co hai so nguyen to Wolstenholme đuoc biet la 16843 va 2124679 (day so A088164  trong bang OEIS). Khong co so nguyen to Wolstenholme nao khac nam duoi 109.


Đinh nghia So nguyen to Wolstenholme co the đinh nghia duoi cac cach sau.


= Đinh nghia qua he so nhi thuc =
So nguyen to Wolstenholme la so nguyen to p > 7 thoa man bieu thuc đong du sau:

  
    
      
        
          
            
              (
            
            
              
                2
                p
                −
                1
              
              
                p
                −
                1
              
            
            
              )
            
          
        
        ≡
        1
        
          
          (
          mod
          
          
            p
            
              4
            
          
          )
        
        ,
      
    
    {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},}
  

Trong đo dau ngoac o ve trai ki hieu he so nhi thuc.
Đinh ly Wolstenholme phat bieu rang voi moi so nguyen to p > 3 quan he đong du sau đuoc thoa man:

  
    
      
        
          
            
              (
            
            
              
                2
                p
                −
                1
              
              
                p
                −
                1
              
            
            
              )
            
          
        
        ≡
        1
        
          
          (
          mod
          
          
            p
            
              3
            
          
          )
        
        .
      
    
    {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.}
  


= Đinh nghia qua cac so Bernoulli =
So nguyen to Wolstenholme la so nguyen to p la uoc cua tu so cua so Bernoulli Bp−3. Do đo cac so nguyen to Wolstenholme tao thanh tap con cua cac so nguyen to phi chinh quy.


= Đinh nghia qua cac cap so phi chinh quy =

So nguyen to Wolstenholme la so nguyen to p sao cho (p, p–3) la cap so phi chinh quy.


= Đinh nghia qua cac so đieu hoa =
So nguyen to Wolstenholme la so nguyen to  p sao cho

  
    
      
        
          H
          
            p
            −
            1
          
        
        ≡
        0
        
          
          (
          mod
          
          
            p
            
              3
            
          
          )
        
        
        ,
      
    
    {\displaystyle H_{p-1}\equiv 0{\pmod {p^{3}}}\,,}
  


Tim kiem so nguyen to Wolstenholme Cong cuoc tim kiem cac so nguyen to Wolstenholme bat đau tu nhung nam 1960 cho toi ngay nay, voi ket qua moi nhat vao nam 2007. So nguyen to Wolstenholme đau tien 16843 đuoc tim thay vao nam 1964, mac du no khong đuoc bao cao vao thoi gian đo. Phat hien nam 1964 sau đo đuoc cong nhan vao nhung nam 1970. Đay la so nguyen to Wolstenholme duy nhat đuoc biet trong suot 20 nam, cho đen khi thong bao tim đuoc them so nguyen to Wolstenholme thu hai 2124679 trong 1993. Cho toi 1,2×107, khong co so nguyen to Wolstenholme nao khac đuoc tim thay. Sau nay đuoc cai tien len toi 2×108 boi McIntosh vao 1995. Trevisan & Weber cai tien them len toi 2,5×108. Ket qua moi nhat trong 2007 bao cao rang khong co so nguyen to Wolstenholme nao khac nam duoi 109.


Phong đoan so so nguyen to Wolstenholme Hien cac nha toan hoc đang gia thuyet rang co vo so so nguyen to Wolstenholme. Hon nua con co gia thuyet so cac so nguyen to Wolstenholme nho hon hoac bang x nam vao khoang ln ln x, trong đo ln ky hieu logarit tu nhien. Voi moi so nguyen to p ≥ 5, thuong Wolstenholme đuoc đinh nghia nhu sau

  
    
      
        
          W
          
            p
          
        
        
          =
        
        
          
            
              
                
                  
                    (
                  
                  
                    
                      2
                      p
                      −
                      1
                    
                    
                      p
                      −
                      1
                    
                  
                  
                    )
                  
                
              
              −
              1
            
            
              p
              
                3
              
            
          
        
        .
      
    
    {\displaystyle W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.}
  

p la so nguyen to Wolstenholme khi va chi khi Wp ≡ 0 (mod p). 


Xem them So nguyen to Wieferich
So nguyen to Wall–Sun–Sun
So nguyen to Wilson


Chu thich Tham khao Đoc them Lien ket ngoai Caldwell, Chris K. Wolstenholme prime from The Prime Glossary
McIntosh, R. J. Wolstenholme Search Status as of March 2004 e-mail to Paul Zimmermann
Bruck, R. Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients
Conrad, K. The p-adic Growth of Harmonic Sums interesting observation involving the two Wolstenholme primes