Created problem_46 in project_euler

project_euler/problem_46/__init__.py

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+#

project_euler/problem_46/sol1.py

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+"""
+It was proposed by Christian Goldbach that every odd composite number can be
+written as the sum of a prime and twice a square.
+
+9 = 7 + 2 × 12
+15 = 7 + 2 × 22
+21 = 3 + 2 × 32
+25 = 7 + 2 × 32
+27 = 19 + 2 × 22
+33 = 31 + 2 × 12
+
+It turns out that the conjecture was false.
+
+What is the smallest odd composite that cannot be written as the sum of a
+prime and twice a square?
+"""
+
+from typing import List
+
+seive = [True] * 100001
+i = 2
+while i * i <= 100000:
+    if seive[i]:
+        for j in range(i * i, 100001, i):
+            seive[j] = False
+    i += 1
+
+
+def is_prime(n: int) -> bool:
+    """
+    Returns True if n is prime,
+    False otherwise, for 2 <= n <= 100000
+    >>> is_prime(87)
+    False
+    >>> is_prime(23)
+    True
+    >>> is_prime(25363)
+    False
+    """
+    return seive[n]
+
+
+odd_composites = [num for num in range(3, len(seive), 2) if not is_prime(num)]
+
+
+def compute_nums(n: int) -> List[int]:
+    """
+    Returns a list of first n odd composite numbers which do
+    not follow the conjecture.
+    >>> compute_nums(1)
+    [5777]
+    >>> compute_nums(2)
+    [5777, 5993]
+    >>> compute_nums(0)
+    Traceback (most recent call last):
+        ...
+    ValueError: n must be => 0
+    >>> compute_nums("a")
+    Traceback (most recent call last):
+        ...
+    ValueError: n must be an exact integer
+    >>> compute_nums(1.1)
+    Traceback (most recent call last):
+        ...
+    ValueError: n must be an exact integer
+    """
+    if not isinstance(n, int):
+        raise ValueError("n must be an exact integer")
+    if n <= 0:
+        raise ValueError("n must be => 0")
+    list_nums = []
+    for num in range(len(odd_composites)):
+        i = 0
+        while 2 * i * i <= odd_composites[num]:
+            rem = odd_composites[num] - 2 * i * i
+            if is_prime(rem):
+                break
+            i += 1
+        else:
+            list_nums.append(odd_composites[num])
+            if len(list_nums) == n:
+                return list_nums
+
+
+if __name__ == "__main__":
+    print(f"{compute_nums(1) = }")