# Digit fifth powers: Euler Problem 30

Euler problem 30 is another number crunching problem that deals with numbers to the power of five. Two other Euler problems dealt withÂ raising numbers to a power. The previous problem looked at permutations of powers and problem 16 asks for the sum of the digits of $2^{1000}$.

Numberphile has a nice video about a trick to quickly calculate the fifth root of a number that makes you look like a mathematical wizard.

## Euler Problem 30 Definition

Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:

$1634 = 1^4 + 6^4 + 3^4 + 4^4$

$8208 = 8^4 + 2^4 + 0^4 + 8^4$

$9474 = 9^4 + 4^4 + 7^4 + 4^4$

As $1 = 1^4$ is not a sum, it is not included.

The sum of these numbers is $1634 + 8208 + 9474 = 19316$. Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

## Proposed Solution

The problem asks for a brute-force solution but we have a halting problem. How far do we need to go before we can be certain there are no sums of fifth power digits? The highest digit is $9$ and $9^5=59049$, which has five digits. If we then look at $5 \times 9^5=295245$, which has six digits and a good endpoint for the loop. The loop itself cycles through the digits of each number and tests whether the sum of the fifth powers equals the number.

largest <- 6 * 9^5
for (n in 2:largest) {
power.sum <-0
i <- n while (i > 0) {
d <- i %% 10
i <- floor(i / 10)
power.sum <- power.sum + d^5
}
if (power.sum == n) {
print(n)