結果

問題 No.577 Prime Powerful Numbers
コンテスト
ユーザー gew1fw
提出日時 2025-06-12 19:00:41
言語 PyPy3
(7.3.15)
結果
TLE  
実行時間 -
コード長 4,408 bytes
コンパイル時間 210 ms
コンパイル使用メモリ 82,328 KB
実行使用メモリ 130,300 KB
最終ジャッジ日時 2025-06-12 19:00:56
合計ジャッジ時間 6,602 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
sample -- * 1
other TLE * 1 -- * 9
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ソースコード

diff # 

import sys
import math

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

sieve_limit = 10**6
is_prime_sieve, primes_list = sieve(sieve_limit)

def is_prime(n):
    if n <= sieve_limit:
        return is_prime_sieve[n]
    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(y, k):
    if y < 0:
        return 0
    if y == 0:
        return 0
    low = 1
    high = y
    while low <= high:
        mid = (low + high) // 2
        try:
            temp = pow(mid, k)
        except OverflowError:
            high = mid - 1
            continue
        if temp == y:
            return mid
        elif temp < y:
            low = mid + 1
        else:
            high = mid - 1
    return high

def is_prime_power(y):
    if y <= 1:
        return False
    if is_prime(y):
        return True
    max_k = y.bit_length()
    for k in range(max_k, 1, -1):
        if k > y.bit_length():
            continue
        root = integer_nth_root(y, k)
        if root == 0 or root == 1:
            continue
        if pow(root, k) != y:
            continue
        if is_prime(root):
            return True
    return False

def solve():
    Q = int(sys.stdin.readline())
    for _ in range(Q):
        N = int(sys.stdin.readline())
        found = False

        for a in range(3, 61):
            max_p = integer_nth_root(N, a)
            if max_p < 2:
                continue
            for p in range(2, max_p + 1):
                if p <= sieve_limit and not is_prime_sieve[p]:
                    continue
                if p > sieve_limit and not is_prime(p):
                    continue
                x = pow(p, a)
                if x > N:
                    break
                y = N - x
                if y <= 0:
                    continue
                if is_prime_power(y):
                    found = True
                    break
            if found:
                break
        if found:
            print("Yes")
            continue

        for p in primes_list:
            x = p * p
            if x > N:
                break
            y = N - x
            if y <= 0:
                continue
            if is_prime_power(y):
                found = True
                break
        if found:
            print("Yes")
            continue

        for p in primes_list:
            x = p
            if x > N:
                break
            y = N - x
            if y <= 0:
                continue
            if is_prime_power(y):
                found = True
                break
        if found:
            print("Yes")
            continue

        for q in primes_list:
            y = q
            x = N - y
            if x <= 0:
                continue
            if is_prime_power(x):
                found = True
                break
        if found:
            print("Yes")
            continue

        for a in range(1, 61):
            x = pow(2, a)
            if x > N:
                break
            y = N - x
            if y <= 0:
                continue
            if is_prime_power(y):
                found = True
                break
        if found:
            print("Yes")
            continue

        small_primes = [3,5,7,11,13,17,19,23,29,31,37]
        for p in small_primes:
            for a in range(1, 61):
                x = pow(p, a)
                if x > N:
                    break
                y = N - x
                if y <= 0:
                    continue
                if is_prime_power(y):
                    found = True
                    break
            if found:
                break
        if found:
            print("Yes")
            continue

        print("No")

if __name__ == "__main__":
    solve()
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