# Project Euler Solution 77: Prime summations

Project Euler Problem 77: Prime summations continues with number partitions, though we now all summands have to be primes.

We have essentially the same problem as in Solution 76: Counting summations. We are to partition integers into sums of primes. The thing to find out is the smallest number which has over 5,000 partitions in this way.

Using our beloved prime generator from Solution 3: Largest prime factor, we can write an iterator for primes up to that number:

def primes_up_to(number: int) -> int:
for prime in prime_generator():
if prime > number:
break
yield prime

Then we adapt the partition function from the previous problem such that it only takes the primes into account. Otherwise it has the same structure. We also need to make sure that we only allow taking just the number when it is a prime itself. For that we use the accelerated prime test from Solution 58: Spiral primes.

@functools.cache
def partitions(number: int, top: int) -> int:
if number == 0:
return 0
elif number == 1:
return 0
elif number == 2:
return 1
else:
result = sum(
partitions(number - x, min(number - x, x)) for x in primes_up_to(top)
)
if number <= top and is_prime_accelerated(number):
result += 1
return result

The solution then seeds the prime generator with a sufficient number of primes. I've lowered the ceiling after learning what the result is. And then just iterate the numbers until it exceeds the threshold.

def solution() -> int:
prime_generator.__defaults__ = (prime_sieve(72),)
for i in itertools.count(10):
if partitions(i, i) > 5_000:
return i

This runs in 0.017 ms, so the algorithm is clever enough.