Question

I spent a little time hacking an R implementation of the lehmann primality test. The function design I borrowed from http://davidkendal.net/articles/2011/12/lehmann-primality-test

Here is my code:

primeTest <- function(n, iter){
  a <- sample(1:(n-1), 1)
    lehmannTest <- function(y, tries){
    x <- ((y^((n-1)/2)) %% n)
    if (tries == 0) {
      return(TRUE)
            }else{          
      if ((x == 1) | (x == (-1 %% n))){
        lehmannTest(sample(1:(n-1), 1), (tries-1))
      }else{
    return(FALSE)
      }
    }
  }
  lehmannTest(a, iter)
}

primeTest(4, 50) # false
primeTest(3, 50) # true
primeTest(10, 50)# false
primeTest(97, 50) # gives false # SHOULD BE TRUE !!!! WTF

prime_test<-c(2,3,5,7,11,13,17 ,19,23,29,31,37)

for (i in 1:length(prime_test)) {
  print(primeTest(prime_test[i], 50))
}

For small primes it works but as soon as i get around ~30, i get a bad looking message and the function stops working correctly:

2: In lehmannTest(a, iter) : probable complete loss of accuracy in modulus

After some investigating i believe it has to do with floating point conversions. Very large numbers are rounded so that the mod function gives a bad response.

Now the questions.

1. Is this a floating point problem? or in my implementation?
2. Is there a purely R solution or is R just bad at this?

Thanks

Solution:

After the great feedback and a hour reading about modular exponentiation algorithms i have a solution. first it is to make my own modular exponentiation function. The basic idea is that modular multiplication allows you calculate intermediate results. you can calculate the mod after each iteration, thus never getting a giant nasty number that swamps the 16-bit R int.

modexp<-function(a, b, n){
    r = 1
    for (i in 1:b){
        r = (r*a) %% n
    }
    return(r)
}


primeTest <- function(n, iter){
   a <- sample(1:(n-1), 1)
    lehmannTest <- function(y, tries){
      x <- modexp(y, (n-1)/2, n)   
    if (tries == 0) {
      return(TRUE)
            }else{          
      if ((x == 1) | (x == (-1 %% n))){
        lehmannTest(sample(1:(n-1), 1), (tries-1))
        }else{
        return(FALSE)
         }
    }
  }
   if( n < 2 ){
     return(FALSE)
     }else if (n ==2) {
       return(TRUE)
       } else{
         lehmannTest(a, iter)
         }
}

primeTest(4, 50) # false
primeTest(3, 50) # true
primeTest(10, 50)# false
primeTest(97, 50) # NOW IS TRUE !!!!


prime_test<-c(5,7,11,13,17 ,19,23,29,31,37,1009)

for (i in 1:length(prime_test)) {
  print(primeTest(prime_test[i], 50))
}
#ALL TRUE

Answer 1

Of course there is a problem with representing integers. In R integers will be represented correctly up to 2^53 – 1 which is about 9e15. And the term y^((n-1)/2) will exceed that even for small numbers easily. You will have to compute (y^((n-1)/2)) %% n by continually squaring y and taking the modulus. That corresponds to the binary representation of (n-1)/2.

Even the ‘real’ number theory programs do it like that — see Wikipedia’s entry on “modular exponentiation”. That said it should be mentioned that programs like R (or Matlab and other systems for numerical computing) may not be a proper environment for implementing number theory algorithms, probably not even as playing fields with small integers.

Edit: The original package was incorrect
You could utilize the function modpower() in package ‘pracma’ like this:

primeTest <- function(n, iter){
  a <- sample(1:(n-1), 1)
    lehmannTest <- function(y, tries){
    x <- modpower(y, (n-1)/2, n)  # ((y^((n-1)/2)) %% n)
    if (tries == 0) {
      return(TRUE)
            }else{          
      if ((x == 1) | (x == (-1 %% n))){
        lehmannTest(sample(1:(n-1), 1), (tries-1))
      }else{
    return(FALSE)
      }
    }
  }
  lehmannTest(a, iter)
}

The following test is successful as 1009 is the only prime in this set:

prime_test <- seq(1001, 1011, by = 2)
for (i in 1:length(prime_test)) {
    print(primeTest(prime_test[i], 50))
}
# FALSE FALSE FALSE FALSE TRUE  FALSE