結果

問題 No.577 Prime Powerful Numbers
コンテスト
ユーザー lam6er
提出日時 2025-03-20 18:49:56
言語 PyPy3
(7.3.15)
結果
TLE  
実行時間 -
コード長 3,394 bytes
コンパイル時間 142 ms
コンパイル使用メモリ 82,768 KB
実行使用メモリ 129,108 KB
最終ジャッジ日時 2025-03-20 18:52:25
合計ジャッジ時間 6,524 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge4
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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 math

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

primes = sieve(10**6)

def is_prime(n):
    if n < 2:
        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 nth_root(n, b):
    if n == 0:
        return 0
    low = 1
    high = n
    while low <= high:
        mid = (low + high) // 2
        try:
            powered = mid ** b
        except OverflowError:
            powered = float('inf')
        if powered == n:
            return mid
        elif powered < n:
            low = mid + 1
        else:
            high = mid - 1
    return None

def is_prime_power(n):
    if n < 2:
        return False
    max_exp = n.bit_length()
    for b in range(max_exp, 0, -1):
        root = nth_root(n, b)
        if root is None:
            continue
        try:
            if root ** b == n and is_prime(root):
                return True
        except OverflowError:
            continue
    return False

Q = int(input())
for _ in range(Q):
    N = int(input())
    if N < 4:
        print("No")
        continue

    found = False
    max_a = 1
    while (2 ** (max_a + 1)) <= (N - 2):
        max_a += 1
    max_a = min(max_a, 60)

    for a in range(1, max_a + 1):
        x = 2 ** a
        if x > N - 2:
            break
        y = N - x
        if y < 2:
            continue
        if is_prime_power(y):
            print("Yes")
            found = True
            break
    if found:
        continue

    if (N - 2) >= 2:
        y = N - 2
        if is_prime_power(y):
            print("Yes")
            continue

    if (N - 3) >= 2:
        y = N - 3
        if is_prime_power(y):
            print("Yes")
            continue

    for p in primes:
        if p > N - 2:
            break
        y = N - p
        if y < 2:
            continue
        if is_prime_power(y):
            print("Yes")
            found = True
            break
    if found:
        continue

    for p in primes:
        x = p ** 2
        if x > N - 2:
            break
        y = N - x
        if y < 2:
            continue
        if is_prime_power(y):
            print("Yes")
            found = True
            break
    if found:
        continue

    for exponent in range(3, 61):
        max_p = int((N - 2) ** (1.0 / exponent))
        if max_p < 2:
            continue
        for p in primes:
            if p > max_p:
                break
            x = p ** exponent
            if x > N - 2:
                continue
            y = N - x
            if y < 2:
                continue
            if is_prime_power(y):
                print("Yes")
                found = True
                break
        if found:
            break
    if found:
        continue

    print("No")
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