結果

問題 No.577 Prime Powerful Numbers
ユーザー gew1fw
提出日時 2025-06-12 14:00:56
言語 PyPy3
(7.3.15)
結果
TLE  
実行時間 -
コード長 3,565 bytes
コンパイル時間 250 ms
コンパイル使用メモリ 82,304 KB
実行使用メモリ 127,616 KB
最終ジャッジ日時 2025-06-12 14:02:08
合計ジャッジ時間 6,663 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge2
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ファイルパターン 結果
sample -- * 1
other TLE * 1 -- * 9
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ソースコード

diff # 

import sys
import math

def sieve(limit):
    sieve = [True] * (limit + 1)
    sieve[0] = sieve[1] = False
    for i in range(2, int(math.sqrt(limit)) + 1):
        if sieve[i]:
            sieve[i*i : limit+1 : i] = [False] * len(sieve[i*i : limit+1 : i])
    primes = [i for i, is_prime in enumerate(sieve) if is_prime]
    return primes

primes = sieve(10**6)

def is_prime(n):
    if n < 2:
        return False
    for p in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]:
        if n % p == 0:
            return n == p
    d = n - 1
    s = 0
    while d % 2 == 0:
        d //= 2
        s += 1
    for a in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]:
        if a >= n:
            continue
        x = pow(a, d, n)
        if x == 1 or x == n - 1:
            continue
        for _ in range(s - 1):
            x = pow(x, 2, n)
            if x == n - 1:
                break
        else:
            return False
    return True

def integer_nth_root(x, n):
    if x == 0:
        return 0
    low = 1
    high = x
    while low <= high:
        mid = (low + high) // 2
        try:
            temp = mid ** n
        except OverflowError:
            temp = float('inf')
        if temp == x:
            return mid
        elif temp < x:
            low = mid + 1
        else:
            high = mid - 1
    return high

def is_prime_power(x):
    if x < 2:
        return False
    max_e = x.bit_length()
    for e in range(max_e, 0, -1):
        k = integer_nth_root(x, e)
        if k ** e == x:
            if is_prime(k):
                return True
    return False

def solve():
    input = sys.stdin.read().split()
    Q = int(input[0])
    for i in range(1, Q + 1):
        N = int(input[i])
        found = False
        
        # Case 1: p is a prime, rem = N - p is a prime power
        for p in primes:
            if p >= N:
                break
            rem = N - p
            if rem < 2:
                continue
            if is_prime_power(rem):
                found = True
                break
        if not found:
            # Check if N-2 is a prime power (since 2 is a prime)
            if (N - 2) >= 2:
                if is_prime_power(N - 2):
                    found = True
        if found:
            print("Yes")
            continue
        
        # Case 2: both terms are prime powers with exponents >= 2
        for a in range(2, 61):
            p_max = integer_nth_root(N, a)
            if p_max < 2:
                continue
            # Check p_max and nearby values
            for delta in range(100):
                p = p_max - delta
                if p < 2:
                    break
                if not is_prime(p):
                    continue
                pa = p ** a
                if pa > N:
                    continue
                rem = N - pa
                if rem < 2:
                    continue
                if is_prime_power(rem):
                    found = True
                    break
            if found:
                break
            # Check primes from sieve up to p_max
            for p in primes:
                if p > p_max:
                    break
                pa = p ** a
                if pa > N:
                    break
                rem = N - pa
                if rem < 2:
                    continue
                if is_prime_power(rem):
                    found = True
                    break
            if found:
                break
        print("Yes" if found else "No")

if __name__ == "__main__":
    solve()
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