Digit Fifth Powers

Problem Statement

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

• 1634 = 1⁴ + 6⁴ + 3⁴ + 4⁴
• 8208 = 8⁴ + 2⁴ + 0⁴ + 8⁴
• 9474 = 9⁴ + 4⁴ + 7⁴ + 4⁴

As 1 = 1⁴ 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.

Approach

The solution involves:

1. Determining an upper bound for the search (6 × 9⁵ = 354,294)
2. For each number, extracting its digits and computing the sum of fifth powers
3. Checking if the sum equals the original number
4. Accumulating all such numbers (excluding 1)