Fansblog 题目传送门 解题思路 Q! % P = (P-1)!/(P-1)...(Q-1) % P. 因为P是质数,根据威尔逊定理,(P-1)!%P=P-1.所以答案就是(P-1)((P-1)...*(Q-1)的逆元)%P.数据很大,用__int128. 代码如下 #include <bits/stdc++.h> #define INF 0x3f3f3f3f using namespace std; typedef long long ll; inline int read(){ in…

题意 给定一个整数 $P$($10^9 \leq p\leq 1^{14}$),设其前一个质数为 $Q$,求 $Q!  \ \% P$. 分析 暴力...说不定好的板子能过. 根据威尔逊定理,如果 $p$ 为质数,则有 $(p-1)! \equiv p-1(mod \ p)$. $\displaystyle Q! = \frac{(P-1)!}{(Q+1)(Q+2)...(p-1)} \equiv  (p-1)*inv\ (mod \ P)$. 根据素数定理,$\displaystyle \pi…

题意:给定质数p,求q!模p的值,其中q为小于p的最大质数 1e9<=p<=1e14 思路:根据质数密度近似分布可以暴力找q并检查 找到q后根据威尔逊定理: 把q+1到p-1这一段的逆元移过去 #include<bits/stdc++.h> using namespace std; typedef long long ll; typedef unsigned int uint; typedef unsigned long long ull; typedef pair<int,…

Fansblog Time Limit: 2000/2000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others) Total Submission(s): 3170 Accepted Submission(s): 671 Problem Description Farmer John keeps a website called 'FansBlog' .Everyday , there are many people visite…

参考博客:https://blog.csdn.net/birdmanqin/article/details/97750844 题目链接:链接:http://acm.hdu.edu.cn/showproblem.php?pid=6608   威尔逊定理:在初等数论中,威尔逊定理给出了判定一个自然数是否为素数的充分必要条件.即:当且仅当p为素数时:( p -1 )! ≡ -1 ( mod p ),但是由于阶乘是呈爆炸增长的,其结论对于实际操作意义不大. 题意:T组样例.每组样例,给出一个素数P(1e…

Problem Description Farmer John keeps a website called ‘FansBlog’ .Everyday , there are many people visited this blog.One day, he find the visits has reached P , which is a prime number.He thinks it is a interesting fact.And he remembers that the vis…

转载请注明出处: http://www.cnblogs.com/fraud/          ——by fraud Zball in Tina Town Time Limit: 3000/1500 MS (Java/Others)    Memory Limit: 262144/262144 K (Java/Others)Total Submission(s): 219    Accepted Submission(s): 144 Problem Description Tina Town i…

YAPTCHA Time Limit: 10000/5000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others) Total Submission(s): 875    Accepted Submission(s): 458 Problem Description The math department has been having problems lately. Due to immense amount of uns…

传送门 题意: 给出自然数 n,计算出 Sn 的值,其中 [ x ]表示不大于 x 的最大整数. 题解: 根据威尔逊定理,如果 p 为素数,那么 (p-1)! ≡ -1(mod p),即 (p-1)! + 1 = p*q. 令 f(K) =  ①如果 3K+7 为素数,则 (3K+7-1)! ≡ -1(mod 3K+7),即 (3K+6)! = (3K+7)*q -1. 那么表达式 可化简为 [ (3K+7)*q / (3K+7) - 1 / (3K+7) ] = [ q - 1 / (3K+7…

<题目链接> 题目大意: The task that is presented to anyone visiting the start page of the math department is as follows: given a natural n, compute  where [x] denotes the largest integer not greater than x. 给出 t 和n,t代表样例组数,根据给出的n算出上面表达式.(注意:[x]表示,不超过x的最大整数)…

<题目链接> Zball in Tina Town Problem Description Tina Town is a friendly place. People there care about each other.Tina has a ball called zball. Zball is magic. It grows larger every day. On the first day, it becomes 1 time as large as its original siz…

YAPTCHA Time Limit: 10000/5000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)Total Submission(s): 1490    Accepted Submission(s): 811 Problem Description The math department has been having problems lately. Due to immense amount of uns…

题意就是叫你求上述那个公式在不同N下的结果. 思路:很显然的将上述式子换下元另p=3k+7则有 Σ[(p-1)!+1/p-[(p-1)!/p]] 接下来用到一个威尔逊定理,如果p为素数则 ( p -1 )! ≡ -1 ( mod p )    即 (p-1)!+1  为 p的整数倍  因此不难发现[*]里面要么为0,要么为1,为1的情况就是p为素数的情况,然后统计k=1-n中 有多少个3*k+1素数就好了 #include <iostream> #include <cstdio>…

参考博客 HDU-2973 题目 Problem Description The math department has been having problems lately. Due to immense amount of unsolicited automated programs which were crawling across their pages, they decided to put Yet-Another-Public-Turing-Test-to-Tell-Compu…

The math department has been having problems lately. Due to immense amount of unsolicited automated programs which were crawling across their pages, they decided to put Yet-Another-Public-Turing-Test-to-Tell-Computers-and-Humans-Apart on their webpag…

威尔逊定理 在初等数论中,威尔逊定理给出了判定一个自然数是否为素数的充分必要条件.即:当且仅当p为素数时:( p -1 )! ≡ -1 ( mod p ),但是由于阶乘是呈爆炸增长的,其结论对于实际操作意义不大. 充分性 如果“p”不是素数,那么它的正因数必然包含在整数1, 2, 3, 4, … ,p− 1 中,因此gcd((p− 1)!,p) > 1,所以我们不可能得到(p− 1)! ≡ −1 (modp).   必要性   若p是素数,取集合 A={1,2,3,...p -1}; 则A 构成…

YAPTCHA Time Limit: 10000/5000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)Total Submission(s): 1885    Accepted Submission(s): 971   Problem Description The math department has been having problems lately. Due to immense amount of u…

Zball in Tina Town Time Limit: 3000/1500 MS (Java/Others)    Memory Limit: 262144/262144 K (Java/Others) Problem Description Tina Town is a friendly place. People there care about each other. Tina has a ball called zball. Zball is magic. It grows lar…

威尔逊原理.即对于素数p,有(p-1)!=-1( mod p). 首先,将原式变形为[ (3×k+6)! % (3×k+7) + 1] / (3×k+7),所以: 1.3×k+7是素数,结果为1, 2.3×k+7不是素数,则假设(3×k+7)=m1*m2*m3……,可知m1,m2,m3……<=3*k+6,则此时(3×k+6)! % (3×k+7) = 0,所以经过取整,式子的答案为0. #include<cstdio> using namespace std; ],sum[]; void…

Problem Description Farmer John keeps a website called ‘FansBlog’ .Everyday , there are many people visited this blog.One day, he find the visits has reached P , which is a prime number.He thinks it is a interesting fact.And he remembers that the vis…

YAPTCHA Time Limit: 10000/5000 MS (Java/Others) Memory Limit: 32768/32768 K (Java/Others) Total Submission(s): 2041 Accepted Submission(s): 1047 Problem Description The math department has been having problems lately. Due to immense amount of unsolic…

0 写在前面 0.0 前言 由于我太菜了,导致一些东西一学就忘,特开此文来记录下最让我头痛的数学相关问题. 一些引用的文字都注释了原文链接,若侵犯了您的权益,敬请告知:若文章中出现错误,也烦请告知. 该文于 2018.3.31 完成最后一次修改(若有出错的地方,之后也会进行维护).其主要内容限于数论和组合计数类数学相关问题.因为版面原因,其余数学方面的总结会以全新的博文呈现. 感谢你的造访. 0.1 记号说明 由于该文完成的间隔跨度太大,不同时期的内容的写法不严谨,甚至 $LaTeX$ 也有许多…

Tina Town is a friendly place. People there care about each other. Tina has a ball called zball. Zball is magic. It grows larger every day. On the first day, it becomes 11 time as large as its original size. On the second day,it will become 22 times…

(本篇无证明,想要证明的去找度娘)o(*≧▽≦)ツ ----------数论四大定理--------- 数论四大定理: 1.威尔逊定理 2.欧拉定理 3.孙子定理(中国剩余定理) 4.费马小定理 (提示:以后出现(mod p)就表示这个公式是在求余p的条件下成立) 1.威尔逊定理:(PS:威尔逊是个厉害人) 当且仅当p为素数时:( p -1 )! ≡ -1 ( mod p ) 或者这么写( p -1 )! ≡ p-1 ( mod p ) 或者说 若p为质数,则p能被(p-1)!+1整除 在初等数…

费马小定理(Fermat Theory) 是数论中的一个重要定理,其内容为: 假如p是质数,且gcd(a,p)=1,那么 a(p-1)≡1(mod p).即:假如a是整数,p是质数,且a,p互质(即两者只有一个公约数1),那么a的(p-1)次方除以p的余数恒等于1. a^(p-1)%p=1 (其中%为取模操作,且a<p,p为质数) 费马小定理是初等数论四大定理(威尔逊定理,欧拉定理(数论中的欧拉定理),中国剩余定理(又称孙子定理)之一,在初等数论中有着非常广泛和重要的应用.实际上,它是欧拉定理的…

题目链接:http://acm.hdu.edu.cn/showproblem.php?pid=6608 大致题意是比p小的最大素数q,求q!%p的值. 由威尔逊定理开始推: $(p-1)!\equiv-1(mod p)$ $(p-1)!modp\equiv p-1$ $q!*(q+1)*(q+2)*...*(p-1)modp\equiv p-1$ $q!modp=\tfrac{p-1}{(q+1)*(q+2)*...*(p-1)}modp$ 然后只需要求出q就可以了,数量级1e9的判断素数可以用…

http://acm.hdu.edu.cn/showproblem.php?pid=6608 题意:给一个比较大的质数P(1e14以内),求比它小的最大的质数Q(貌似保证存在的样子,反正我没判不存在),求Q!modP的值. 一开始觉得是个什么神仙题,但是怎么全场都绿了一片,瞬时感觉智商受到了侮辱.想了很多什么奇奇怪怪的性质,都不太清楚,然后队友卡1007的时候我打了个对P找ans的表看了一下.发现当P-Q=2的时候就是ans=1??? 然后打个(P-2)!%P的表,发现小数据都这样过,估计也是真…

ACM数论之旅5---数论四大定理(你怕不怕(☆ﾟ∀ﾟ)老实告诉我)   (本篇无证明,想要证明的去找度娘)o(*≧▽≦)ツ ----------数论四大定理--------- 数论四大定理: 1.威尔逊定理 2.欧拉定理 3.孙子定理(中国剩余定理) 4.费马小定理 (提示:以后出现(mod p)就表示这个公式是在求余p的条件下成立) 1.威尔逊定理:(PS:威尔逊是个厉害人) 当且仅当p为素数时:( p -1 )! ≡ -1 ( mod p ) 或者这么写( p -1 )! ≡ p-1 (…

从去年还在竞赛的时候2/12的原博客里搬运来的 不得不说之前取名真的很艺术qwq 今天开始上的数论课,让头发以肉眼可见的速度掉落emmm 没关系我头发多我不怕啦啦啦QwQ 其中最令人头疼的就是那些人名定理了,看到它们,总是在想: 我是谁 我从哪里来 我要做什么 咳咳,不是,是下面-- 这是什么 怎么证明 代码实现 有什么用 其中怎么证明过于复杂了,一点也没听懂.数竞的事情关我们信竞什么事...... 所以这篇博客将整理1/3/4的问题,理清楚欧拉费马欧几里得威尔逊等等名人定理之间的关系![当然没…

数学素养 low,表达可能存在不严谨,见谅.我准备慢慢补上证明? Theorems. 裴蜀定理:关于 \(x, y\) 的线性方程 \(ax + by = c\) 有解,当且仅当 \(\gcd (a, b) | c\). 唯一分解定理:对于任意一个大于 \(1\) 的整数 \(n\),\(n\) 可以唯一地被分解成若干个质数的幂的乘积. 欧拉定理:若 \(\gcd (a, m) = 1\),则 \(a ^{\varphi (m)} \equiv 1 \pmod m\).且可知 \(a ^{n \…