Project Euler Solution 87: Prime Power Triples

In Project Euler Problem 87 we are supposed to find out how many numbers below fifty million can be expressed as a sum of a prime square, a prime cube and a prime to the fourth power.

My approach here is to just iterate through all numbers and see when we reach the ceiling.

First we write a helper function which yields all numbers raised to a fixed exponent:

def powers(primes: list[int], exponent: int) -> Iterator[int]:
    for base in primes:
        yield base**exponent

Then we use the prime sieve from solution 7. With that we go through all the squares of prime numbers until we hit the ceiling. Then we add all the cubes, again watching the ceiling. Also with the fourth powers. If the sum of these four summands is above the ceiling, we stop there. Otherwise we just add the number to the set of numbers which can be represented like this.

def solution() -> int:
    ceiling = 50_000_000
    primes = prime_sieve(int(math.sqrt(ceiling)))
    numbers = set()
    for square in powers(primes, 2):
        if square > ceiling:
        for cube in powers(primes, 3):
            if square + cube > ceiling:
            for quadruple in powers(primes, 4):
                number = square + cube + quadruple
                if number >= ceiling:
    return len(numbers)

This way we don't have to go through all numbers up to 50 million and try to decompose them. Rather we can just construct all the numbers that can be written like this.

This runs in 481 ms, so it's not instant. But it should be good enough.