Project Euler Problem #3 (R)

The third problem in Project Euler is an interesting one:

The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143 ?

In order to compute prime factors, we can use a sieving method. The simplest and oldest sieving algorithm is the sieve of Erastothenes, which we can implement in R as follows:

sieve <- function(n) {
    if (n <= 1) {
        return(numeric(0))
    }
    if (n == 2) {
        return(c(2))
    }

    i <- 1
    while (length(a) > 0) {
        b[i] <- a[1]
        i <- i + 1
        a <- a[!(a %% a[1] == 0)]
    }
    b
}

So now we need to compute the largest prime factor of 600851475143. An upper bound for the factorisation is \(p \leq \lfloor\sqrt{x}\rfloor\), so we can halt our search at this point (alternatively, we may be able to begin our search here and work downwards). Let’s do it the obvious (and inefficient) way, starting at 1 and working up to \(\lfloor\sqrt{x}\rfloor\), building up a list of prime factors along the way:

pfactor <- function(x) {
    factors <- c()
    j <- 1
    primes <- sieve( floor(sqrt(x)) )
    for (prime in primes) {
        if (x %% prime == 0) {
            factors[j] <- prime
            j <- j + 1
    }
    }
    factors
}

Now to compute the maximum factor, we just take the maximum of this list:

max( pfactor(600851475143) )