**Problem:** Find the sum of all the primes below two million.

**Commentary:** If you’re not careful, this problem can really get you
in R. In poking around the internet, there are a few solutions to this
problem, but all the ones I’ve tested are slow. Some, even from people
who are better at R than me, are VERY
slow. That linked
one in particular took 6 minutes and 8.728000 seconds to complete. My
solutions runs in under 2 seconds.

So we’ll be using that old prime algorithm I’ve already discussed to death. Other than a few tricks in there that I never see get implemented, there’s a crucial trick to this solution. Since we only want the sum, let’s not be slow about finding the primes we want. I only have to find all primes below , where . It takes basically no time to find them. From there, I start evaluating my sum, and like a good little R user, I vectorize by getting the possible list of possible primes above . From there, I check if any of my known primes divide into those possible primes. This is actually much faster than generating primes and storing them.** **

**R Code:**

```
ElapsedTime <- system.time({
##########################
n <- 2000000
### Function to get primes below specified n; option to get only those below sqrt(n)
PrimesBelow <- function(n, below.sqrt=FALSE){
if (below.sqrt == TRUE){
m <- ceiling(sqrt(n))
} else {
m <- n
}
primes <- c(2,3)
i <- 3
while (i < m-1){
flag <- 0
i <- i+2
if (i%%6 == 3){
flag <- 1
}
if (flag == 0){
s <- sqrt(i)+1
possible.primes <- primes[primes<s]
for (prime in possible.primes){
if ((i%%prime == 0)){
flag <- 1
break
}
}
if (flag == 0){
primes <- c(primes, i)
}
}
}
primes
}
###
primes <- PrimesBelow(n, below.sqrt=TRUE)
biglist <- ceiling(sqrt(n)):n
biglist <- biglist[-which(biglist%%2 == 0)]
biglist <- biglist[-which(biglist%%6 == 3)]
for (prime in primes){
biglist <- biglist[which(biglist%%prime != 0)]
}
answer <- sum(primes)+sum(as.numeric(biglist))
##########################
})[3]
ElapsedMins <- floor(ElapsedTime/60)
ElapsedSecs <- (ElapsedTime-ElapsedMins*60)
cat(sprintf("\nThe answer is: %f\nTotal elapsed time: %d minutes and %f seconds\n", answer, ElapsedMins, ElapsedSecs))
```

**Output:**

```
The answer is: 142913828922.000000
Total elapsed time: 0 minutes and 1.915000 seconds
```