Project Euler Problem 30

Solution

Answer: 443839

We seek all integers that can be written as the sum of the fifth powers of their digits.

1. Mathematical analysis

We first determine an upper bound.

For an $n$-digit number, the maximum possible sum of fifth powers of its digits is:

$$n \cdot 9^5$$

Since:

$$9^5 = 59049$$

Check when an $n$-digit number can no longer equal such a sum:

$6 \cdot 9^5 = 354294$
A 7-digit number is at least $1{,}000{,}000$

But:

$$7 \cdot 9^5 = 413343 < 1{,}000{,}000$$

So no 7-digit number can satisfy the condition. Therefore we only need to search up to:

$$354294$$

excluding $1$, since it is not considered a sum.

2. Clean Python code

def digit_fifth_power_sum():
    limit = 6 * (9 ** 5)  # 354294
    total = 0

    for n in range(2, limit + 1):
        if n == sum(int(d) ** 5 for d in str(n)):
            total += n

    return total

print(digit_fifth_power_sum())

3. Mental trace / correctness check

The numbers that satisfy the property are:

4150
4151
54748
92727
93084
194979

Check one example:

$$4150 = 4^5 + 1^5 + 5^5 + 0^5$$

$$= 1024 + 1 + 3125 + 0 = 4150$$

Summing them:

$$4150 + 4151 + 54748 + 92727 + 93084 + 194979 = 443839$$

Answer: 443839