Added solution for Project Euler problem 87. Fixes: #2695

Author: fpringle

project_euler/problem_87/__init__.py

@@ -0,0 +1,50 @@
+"""
+The smallest number expressible as the sum of a prime square, prime cube, and prime
+fourth power is 28. In fact, there are exactly four numbers below fifty that can be
+expressed in such a way:
+
+28 = 22 + 23 + 24
+33 = 32 + 23 + 24
+49 = 52 + 23 + 24
+47 = 22 + 33 + 24
+
+How many numbers below fifty million can be expressed as the sum of a prime square,
+prime cube, and prime fourth power?
+"""
+
+
+def solution(limit: int = 50000000) -> int:
+    """
+    Return the number of integers less than limit which can be expressed as the sum
+    of a prime square, prime cube, and prime fourth power.
+    >>> solution(50)
+    4
+    """
+    ret = set()
+    prime_square_limit = int((limit - 24) ** (1 / 2))
+
+    primes = set(range(3, prime_square_limit + 1, 2))
+    primes.add(2)
+    for p in range(3, prime_square_limit + 1, 2):
+        if p not in primes:
+            continue
+        primes.difference_update(set(range(p * p, prime_square_limit + 1, p)))
+
+    for prime1 in primes:
+        square = prime1 * prime1
+        for prime2 in primes:
+            cube = prime2 * prime2 * prime2
+            if square + cube >= limit - 16:
+                break
+            for prime3 in primes:
+                tetr = prime3 * prime3 * prime3 * prime3
+                total = square + cube + tetr
+                if total >= limit:
+                    break
+                ret.add(total)
+
+    return len(ret)
+
+
+if __name__ == "__main__":
+    print(solution())