結果

問題 No.577 Prime Powerful Numbers
ユーザー gew1fw
提出日時 2025-06-12 19:01:12
言語 PyPy3
(7.3.15)
結果
TLE  
実行時間 -
コード長 3,849 bytes
コンパイル時間 195 ms
コンパイル使用メモリ 82,460 KB
実行使用メモリ 139,688 KB
最終ジャッジ日時 2025-06-12 19:01:34
合計ジャッジ時間 6,582 ms
ジャッジサーバーID
（参考情報） 		judge4 / judge5
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ファイルパターン 結果
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other TLE * 1 -- * 9
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ソースコード

diff # 

import math
import sys

def is_prime(n):
    if n <= 1:
        return False
    elif n <= 3:
        return True
    elif n % 2 == 0:
        return False
    d = n - 1
    s = 0
    while d % 2 == 0:
        d //= 2
        s += 1
    bases = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
    for a in bases:
        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 is_prime_power(n):
    if is_prime(n):
        return True
    max_b = int(math.log2(n)) + 1
    for b in range(2, max_b + 1):
        low = 1
        high = n
        found = False
        while low <= high:
            mid = (low + high) // 2
            power = mid ** b
            if power == n:
                found = True
                break
            elif power < n:
                low = mid + 1
            else:
                high = mid - 1
        if found and is_prime(mid):
            return True
    return False

def generate_high_powers(N):
    candidates = []
    p = 2
    max_p = int(math.isqrt(N)) + 1
    while p <= max_p:
        if is_prime(p):
            a = 2
            current = p * p
            while current <= N:
                candidates.append(current)
                a += 1
                next_val = p ** a
                if next_val > N or next_val < current:
                    break
                current = next_val
        p += 1
        if p > 10**6:
            break
    return candidates

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]:
            for j in range(i*i, limit+1, i):
                sieve[j] = False
    primes = [i for i, val in enumerate(sieve) if val]
    return primes

def generate_low_primes(N):
    limit = 10**6
    primes = sieve(limit)
    return [p for p in primes if p <= N]

def generate_near_primes(N):
    candidates = []
    max_qb = 10**6
    qb_list = set()
    primes = sieve(int(math.isqrt(max_qb)) + 1)
    for q in primes:
        if q < 2:
            continue
        a = 2
        while True:
            q_power = q ** a
            if q_power > max_qb:
                break
            qb_list.add(q_power)
            a += 1
    for q in primes:
        if q <= max_qb:
            qb_list.add(q)
    for qb in qb_list:
        x = N - qb
        if x >= 2 and is_prime(x):
            candidates.append(x)
    for delta in [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]:
        x = N - delta
        if x >= 2 and is_prime(x):
            candidates.append(x)
    return candidates

def solve():
    Q = int(sys.stdin.readline())
    for _ in range(Q):
        N = int(sys.stdin.readline())
        if N < 2:
            print("No")
            continue
        high_powers = generate_high_powers(N)
        low_primes = generate_low_primes(N)
        near_primes = generate_near_primes(N)
        candidates = set(high_powers + low_primes + near_primes)
        found = False
        for x in candidates:
            if x > N:
                continue
            y = N - x
            if y < 1:
                continue
            if is_prime_power(y):
                found = True
                break
        if not found:
            for x in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]:
                if x > N:
                    continue
                y = N - x
                if y < 1:
                    continue
                if is_prime_power(y):
                    found = True
                    break
        if found:
            print("Yes")
        else:
            print("No")

if __name__ == "__main__":
    solve()
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