project euler – Problem 44

November 8, 2011

(This article was first published on YGC » R, and kindly contributed to R-bloggers)

Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...

It can be seen that P4 + P7 = 22 + 70 = 92 = P8. However, their difference, 70 − 22 = 48, is not pentagonal.

Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference is pentagonal and D = |Pk − Pj| is minimised; what is the value of D?

Calculate differences directly and test the differences to be pentagonal, if so test the sum to be pentagonal.
Use a while loop for have no idea the upper limit should be.

is.pent <- function(x) {
    n <- (1 + sqrt(24*x+1))/6
    if (n == floor(n)) {
    } else {

n <- 2
flag <- TRUE
pent <- 1
while(flag) {
    p <- n*(3*n-1)/2
    diff <- p-pent
    for (i in 1:length(diff)) {
        if(is.pent(diff[i])) {
            s <- p+pent[i]
            if (is.pent(s)) {
                flag <- FALSE
    pent <- c(pent,p)
    n <- n+1
> system.time(source("Problem44.R"))
[1] 5482660
   user  system elapsed 
  12.64    0.00   12.67 

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