Generalized inversive congruential pseudorandom numbers

An approach to nonlinear congruential methods of <a href="/facts/Pseudorandom_number_generator/sIrCxIBq">generating uniform pseudorandom numbers</a> in the interval [0,1) is the <a href="/facts/Inversive_congruential_generator/bRNhg1hA">Inversive congruential generator</a> with prime modulus. A generalization for arbitrary composite moduli 
 
 
 
 m
 =
 
 p
 
 1
 
 
 ,
 …
 
 p
 
 r
 
 
 
 
 {\displaystyle m=p_{1},\dots p_{r}}
 
 with arbitrary distinct <a href="/facts/Prime_number/L20FLkgn">primes</a> 
 
 
 
 
 p
 
 1
 
 
 ,
 …
 ,
 
 p
 
 r
 
 
 ≥
 5
 
 
 {\displaystyle p_{1},\dots ,p_{r}\geq 5}
 
 will be present here.
Let 
 
 
 
 
 
 Z
 
 
 m
 
 
 =
 {
 0
 ,
 1
 ,
 .
 .
 .
 ,
 m
 −
 1
 }
 
 
 {\displaystyle \mathbb {Z} _{m}=\{0,1,...,m-1\}}
 
. For <a href="/facts/Integer/PUwwrawx">integers</a> 
 
 
 
 a
 ,
 b
 ∈
 
 
 Z
 
 
 m
 
 
 
 
 {\displaystyle a,b\in \mathbb {Z} _{m}}
 
 with gcd (a,m) = 1 a generalized inversive congruential sequence 
 
 
 
 (
 
 y
 
 n
 
 
 
 )
 
 n
 ⩾
 0
 
 
 
 
 {\displaystyle (y_{n})_{n\geqslant 0}}
 
 of elements of 
 
 
 
 
 
 Z
 
 
 m
 
 
 
 
 {\displaystyle \mathbb {Z} _{m}}
 
 is defined by

y
          
            0
          
        
        =
        
          
            s
            e
            e
            d
          
        
      
    
    {\displaystyle y_{0}={\rm {seed}}}

y
          
            n
            +
            1
          
        
        ≡
        a
        
          y
          
            n
          
          
            φ
            (
            m
            )
            −
            1
          
        
        +
        b
        
          
          (
          mod
          
          m
          )
        
        
          , 
        
        n
        ⩾
        0
      
    
    {\displaystyle y_{n+1}\equiv ay_{n}^{\varphi (m)-1}+b{\pmod {m}}{\text{,  }}n\geqslant 0}

where 
 
 
 
 φ
 (
 m
 )
 =
 (
 
 p
 
 1
 
 
 −
 1
 )
 …
 (
 
 p
 
 r
 
 
 −
 1
 )
 
 
 {\displaystyle \varphi (m)=(p_{1}-1)\dots (p_{r}-1)}
 
 denotes the number of positive integers less than m which are <a href="/facts/Coprime/bnCxCXxs">relatively prime</a> to m.

Generalized inversive congruential pseudorandom numbers open-in-new

Generalized inversive congruential pseudorandom numbers