数论及其应用——数论基础

  同其他专题一样，在数论专题当中我也开了一篇“基础”文章，其实严谨的来说也不能算基础，其目的在于记录一些有关数论方面的一些理论性或者说模型化不是很强的问题。例如找规律、简单的模拟题之类。

 

  那我们们来看这要一道问题。(Problem source : pku 2183)

 

Description

Farmer John loves to help the cows further their mathematical skills. He has promised them Hay-flavored ice cream if they can solve various mathematical problems.
He said to Bessie, "Choose a six digit integer, and tell me what it is. Then extract the middle four digits. Square them and discard digits at the top until you have another number six digits or shorter. Tell me the result."
Bessie, a mathematical genius in disguise, chose the six digit number 655554. "Moo: 6 5 5 5 5 4", she said. She then extracted the middle four digits:  5555 and squared them: 30858025. She kept only the bottom six digits:  858025. "Moo: 8 5 8 0 2 5", she replied to FJ.
FJ nodded wisely, acknowledging Bessie's prowess in arithmetic.  "Now keep doing that until you encounter a number that repeats a number already seen," he requested.
Bessie decided she'd better create a table to keep everything straight:

              Middle    Middle   Shrunk to
        Num   4 digits   square   6 or fewer
      655554    5555    30858025    858025
      858025    5802    33663204    663204
      663204    6320    39942400    942400
      942400    4240    17977600    977600
      977600    7760    60217600    217600 <-+
      217600    1760     3097600     97600   |
       97600    9760    95257600    257600   |
      257600    5760    33177600    177600   |
      177600    7760    60217600    217600 --+

Bessie showed her table to FJ who smiled and produced a big dish of delicious hay ice cream.  "That's right, Bessie," he praised. "The chain repeats in a loop of four numbers, of which the first encountered was 217600.  The loop was detected after nine iterations."
Help the other cows win ice cream treats.  Given a six digit number, calculate the total number of iterations to detect a loop, the first looping number encountered, and also the length of the loop.
FJ wondered if Bessie knew all the tricks. He had made a table to help her, but she never asked:

              Middle    Middle   Shrunk to
        Num   4 digits   square   6 or fewer
      200023    0002       4        4
           4       0       0        0
           0       0       0        0         [a self-loop]

whose results would be: three iterations to detect a loop, looping on 0, and a length of loop equal to 1.
Remember: Your program can use no more than 16MB of memory.

 

  题目大意：给出一个六位数，取出中间四位数，进行平方运算，然后取出结果的后六位，然后重复上述操作，直到出现了相同的六位数停止。需要你输出这之前进行了多少次操作，并且输出这个第二次出现的六位数，以及该数出现第一次而第二次中间发生的操作数，从问题描述的图示我们直观的理解其含义。

  编程实现：数理分析上几乎不存在难度，编程实现也只是需要简单的模型实现。

  参考代码如下。

 

#include<stdlib.h>
#include<stdio.h>
#include<string.h>
#include<iostream>
using namespace std;
int use[1000001];

int main()
{
     int x , y, i;
     scanf("%d",&x);
     memset(use , -1 , sizeof(use));
     use[x] = 0;
     for(i = 1;;i++)
     {
       x = x / 10 % 10000;
       x *= x;
       x %= 1000000;
       if(use[x] >= 0)  break;
       use[x] = i;
     }

     printf("%d %d %d\n",x,i-use[x],i);
}