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53 lines
1.5 KiB
Python
53 lines
1.5 KiB
Python
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"""
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Project Euler Problem 87: https://projecteuler.net/problem=87
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The smallest number expressible as the sum of a prime square, prime cube, and prime
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fourth power is 28. In fact, there are exactly four numbers below fifty that can be
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expressed in such a way:
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28 = 22 + 23 + 24
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33 = 32 + 23 + 24
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49 = 52 + 23 + 24
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47 = 22 + 33 + 24
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How many numbers below fifty million can be expressed as the sum of a prime square,
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prime cube, and prime fourth power?
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"""
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def solution(limit: int = 50000000) -> int:
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"""
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Return the number of integers less than limit which can be expressed as the sum
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of a prime square, prime cube, and prime fourth power.
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>>> solution(50)
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4
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"""
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ret = set()
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prime_square_limit = int((limit - 24) ** (1 / 2))
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primes = set(range(3, prime_square_limit + 1, 2))
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primes.add(2)
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for p in range(3, prime_square_limit + 1, 2):
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if p not in primes:
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continue
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primes.difference_update(set(range(p * p, prime_square_limit + 1, p)))
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for prime1 in primes:
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square = prime1 * prime1
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for prime2 in primes:
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cube = prime2 * prime2 * prime2
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if square + cube >= limit - 16:
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break
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for prime3 in primes:
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tetr = prime3 * prime3 * prime3 * prime3
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total = square + cube + tetr
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if total >= limit:
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break
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ret.add(total)
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return len(ret)
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if __name__ == "__main__":
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print(f"{solution() = }")
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