P1217 [USACO1.5]回文质数 Prime Palindromes（质数 回文数）

题的链接：P1217 [USACO1.5]回文质数 Prime Palindromes

知识点：偶数位数回文数（除11）必定不是质数（自行百度）

打表流程： 暴力输出最大范围内所有答案，按空格隔开，最后用软件将空格全部替换为‘，’即可；再按条件输出。这里用到了线性筛法，没必要，用最笨的办法就行！

参考代码1.0： 最后一个数据点超时，直接打表暴力通过AC，哈哈哈！

#include <queue>
#include <cstdio>
#include <string>
#include <cstring>
#include <iostream>
#include <algorithm>
#define INF 0x3f3f3f3f
#define MAX 20010
using namespace std;

int a, b;
int primes[100000010], cnt;
int res[100000010], nes;
int pos;
int S[100010] = {2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,
797,919,929,10301,10501,10601,11311,11411,12421,12721,12821,13331,13831,13931,
14341,14741,15451,15551,16061,16361,16561,16661,17471,17971,18181,18481,19391,
19891,19991,30103,30203,30403,30703,30803,31013,31513,32323,32423,33533,34543,
34843,35053,35153,35353,35753,36263,36563,37273,37573,38083,38183,38783,39293,
70207,70507,70607,71317,71917,72227,72727,73037,73237,73637,74047,74747,75557,
76367,76667,77377,77477,77977,78487,78787,78887,79397,79697,79997,90709,91019,
93139,93239,93739,94049,94349,94649,94849,94949,95959,96269,96469,96769,97379,
97579,97879,98389,98689,1003001,1008001,1022201,1028201,1035301,1043401,1055501,
1062601,1065601,1074701,1082801,1085801,1092901,1093901,1114111,1117111,1120211,
1123211,1126211,1129211,1134311,1145411,1150511,1153511,1160611,1163611,1175711,
1177711,1178711,1180811,1183811,1186811,1190911,1193911,1196911,1201021,1208021,
1212121,1215121,1218121,1221221,1235321,1242421,1243421,1245421,1250521,1253521,
1257521,1262621,1268621,1273721,1276721,1278721,1280821,1281821,1286821,1287821,
1300031,1303031,1311131,1317131,1327231,1328231,1333331,1335331,1338331,1343431,
1360631,1362631,1363631,1371731,1374731,1390931,1407041,1409041,1411141,1412141,
1422241,1437341,1444441,1447441,1452541,1456541,1461641,1463641,1464641,1469641,
1486841,1489841,1490941,1496941,1508051,1513151,1520251,1532351,1535351,1542451,
1548451,1550551,1551551,1556551,1557551,1565651,1572751,1579751,1580851,1583851,
1589851,1594951,1597951,1598951,1600061,1609061,1611161,1616161,1628261,1630361,
1633361,1640461,1643461,1646461,1654561,1657561,1658561,1660661,1670761,1684861,
1685861,1688861,1695961,1703071,1707071,1712171,1714171,1730371,1734371,1737371,
1748471,1755571,1761671,1764671,1777771,1793971,1802081,1805081,1820281,1823281,
1824281,1826281,1829281,1831381,1832381,1842481,1851581,1853581,1856581,1865681,
1876781,1878781,1879781,1880881,1881881,1883881,1884881,1895981,1903091,1908091,
1909091,1917191,1924291,1930391,1936391,1941491,1951591,1952591,1957591,1958591,
1963691,1968691,1969691,1970791,1976791,1981891,1982891,1984891,1987891,1988891,
1993991,1995991,1998991,3001003,3002003,3007003,3016103,3026203,3064603,3065603,
3072703,3073703,3075703,3083803,3089803,3091903,3095903,3103013,3106013,3127213,
3135313,3140413,3155513,3158513,3160613,3166613,3181813,3187813,3193913,3196913,
3198913,3211123,3212123,3218123,3222223,3223223,3228223,3233323,3236323,3241423,
3245423,3252523,3256523,3258523,3260623,3267623,3272723,3283823,3285823,3286823,
3288823,3291923,3293923,3304033,3305033,3307033,3310133,3315133,3319133,3321233,
3329233,3331333,3337333,3343433,3353533,3362633,3364633,3365633,3368633,3380833,
3391933,3392933,3400043,3411143,3417143,3424243,3425243,3427243,3439343,3441443,
3443443,3444443,3447443,3449443,3452543,3460643,3466643,3470743,3479743,3485843,
3487843,3503053,3515153,3517153,3528253,3541453,3553553,3558553,3563653,3569653,
3586853,3589853,3590953,3591953,3594953,3601063,3607063,3618163,3621263,3627263,
3635363,3643463,3646463,3670763,3673763,3680863,3689863,3698963,3708073,3709073,
3716173,3717173,3721273,3722273,3728273,3732373,3743473,3746473,3762673,3763673,
3765673,3768673,3769673,3773773,3774773,3781873,3784873,3792973,3793973,3799973,
3804083,3806083,3812183,3814183,3826283,3829283,3836383,3842483,3853583,3858583,
3863683,3864683,3867683,3869683,3871783,3878783,3893983,3899983,3913193,3916193,
3918193,3924293,3927293,3931393,3938393,3942493,3946493,3948493,3964693,3970793,
3983893,3991993,3994993,3997993,3998993,7014107,7035307,7036307,7041407,7046407,
7057507,7065607,7069607,7073707,7079707,7082807,7084807,7087807,7093907,7096907,
7100017,7114117,7115117,7118117,7129217,7134317,7136317,7141417,7145417,7155517,
7156517,7158517,7159517,7177717,7190917,7194917,7215127,7226227,7246427,7249427,
7250527,7256527,7257527,7261627,7267627,7276727,7278727,7291927,7300037,7302037,
7310137,7314137,7324237,7327237,7347437,7352537,7354537,7362637,7365637,7381837,
7388837,7392937,7401047,7403047,7409047,7415147,7434347,7436347,7439347,7452547,
7461647,7466647,7472747,7475747,7485847,7486847,7489847,7493947,7507057,7508057,
7518157,7519157,7521257,7527257,7540457,7562657,7564657,7576757,7586857,7592957,
7594957,7600067,7611167,7619167,7622267,7630367,7632367,7644467,7654567,7662667,
7665667,7666667,7668667,7669667,7674767,7681867,7690967,7693967,7696967,7715177,
7718177,7722277,7729277,7733377,7742477,7747477,7750577,7758577,7764677,7772777,
7774777,7778777,7782877,7783877,7791977,7794977,7807087,7819187,7820287,7821287,
7831387,7832387,7838387,7843487,7850587,7856587,7865687,7867687,7868687,7873787,
7884887,7891987,7897987,7913197,7916197,7930397,7933397,7935397,7938397,7941497,
7943497,7949497,7957597,7958597,7960697,7977797,7984897,7985897,7987897,7996997,
9002009,9015109,9024209,9037309,9042409,9043409,9045409,9046409,9049409,9067609,
9073709,9076709,9078709,9091909,9095909,9103019,9109019,9110119,9127219,9128219,
9136319,9149419,9169619,9173719,9174719,9179719,9185819,9196919,9199919,9200029,
9209029,9212129,9217129,9222229,9223229,9230329,9231329,9255529,9269629,9271729,
9277729,9280829,9286829,9289829,9318139,9320239,9324239,9329239,9332339,9338339,
9351539,9357539,9375739,9384839,9397939,9400049,9414149,9419149,9433349,9439349,
9440449,9446449,9451549,9470749,9477749,9492949,9493949,9495949,9504059,9514159,
9526259,9529259,9547459,9556559,9558559,9561659,9577759,9583859,9585859,9586859,
9601069,9602069,9604069,9610169,9620269,9624269,9626269,9632369,9634369,9645469,
9650569,9657569,9670769,9686869,9700079,9709079,9711179,9714179,9724279,9727279,
9732379,9733379,9743479,9749479,9752579,9754579,9758579,9762679,9770779,9776779,
9779779,9781879,9782879,9787879,9788879,9795979,9801089,9807089,9809089,9817189,
9818189,9820289,9822289,9836389,9837389,9845489,9852589,9871789,9888889,9889889,
9896989,9902099,9907099,9908099,9916199,9918199,9919199,9921299,9923299,9926299,
9927299,9931399,9932399,9935399,9938399,9957599,9965699,9978799,9980899,9981899,
9989899};
bool vis[100000010];

void get_primes(int N)
{
    for(int i = 2; i <= N; i++)
    {
        if(!vis[i]) primes[cnt++] = i;
        for(int j = 0; primes[j] <= N / i; j++)
        {
            vis[primes[j] * i] = true;
            if(i % primes[j] == 0) break;
        }
    }
}

void get_huiwen()
{
    for(int i = 0; i < cnt; i++)
    {
        int t = primes[i], tem = 0;
        if(t < a) continue;
        while(t)
        {
            tem = tem*10 + t%10;
            t /= 10;
        }
        if(primes[i] == tem) cout << tem <<" ", pos++;
    }
}

int main()
{
    cin >> a >> b;
    //打表存储直接输出。。。
    //get_primes(b);
    //get_huiwen();
    //cout <<endl<< pos << endl;
    for(int i = 0; i < 1000; i++)
    {
        if(S[i] >= a && S[i] <= b) cout << S[i] << endl;
        if(S[i] > b) break;
    }
    return 0;
}

参考代码2.0： 朴素筛法和线性筛法都可以通过，直接在main函数里将输入的b > 10000000的直接改为10000000，因为（偶数位数回文数（除11）必定不是质数）；这里不改的话，最后一个测试点超时，通不过。

#include <queue>
#include <cstdio>
#include <string>
#include <cstring>
#include <iostream>
#include <algorithm>
#define INF 0x3f3f3f3f
#define MAX 20010
using namespace std;

int a, b;
int primes[10000010], cnt;
int res[10000010], nes;
bool vis[10000010];
//偶数位数回文数（除11）必定不是质数,所以计算到10000000即可


//线性筛法求素数
void get_primes1(int N)
{
    for(int i = 2; i <= N; i++)
    {
        if(!vis[i]) primes[cnt++] = i;
        for(int j = 0; primes[j] <= N / i; j++)
        {
            vis[primes[j] * i] = true;
            if(i % primes[j] == 0) break;
        }
    }
}

//朴素筛法求素数
void get_primes2(int N)
{
    for(int i = 2; i <= N; i++)
    {
        if(vis[i]) continue;
        primes[cnt++] = i;
        for(int j = i; j <= N; j += i) vis[j] = true;
    }
}

void get_huiwen()
{
    for(int i = 0; i < cnt; i++)
    {
        int t = primes[i], tem = 0;
        if(t < a) continue;
        while(t)
        {
            tem = tem*10 + t%10;
            t /= 10;
        }
        if(primes[i] == tem) cout << tem << endl;
    }
}

int main()
{
    cin >> a >> b;
    //关键点
    if(b > 10000000) b = 10000000;
    get_primes1(b);
    get_huiwen();
    return 0;
}

参考代码3.0： 回文数肯定比质数少，所以DFS构造回文数，再判断构造的范围，小于a则return，大于b则直接退出本次dfs；范围内的质数则输出；

回文数构造： 按照位数构造，由于是回文数，所以中间的前后都是对称的，eg：12321,345543；所以可以dfs来填前半部分的值，后半部分的值for循环来复制，而中间值的判断为：(k+1)/2，eg: 5，6的中间都为3，dfs前三位数就可以构造5到6位的回文数，然后将数组存的数加工成整数进行判断；

关键部分注意：

• 大于b的时候，将vis标记为false，进行下一次循环的退出操作；
• 判断当前位数时，若为0即首位，首位不能为0，所以特判一下；
• 偶数回文数肯定不是质数（除了11），所以主函数特判一下，以及9位数及一亿不是质数特判一下；
• 主函数dfs之前一定要将vis重新更新为true
• 计算数字位数时直接调用cmath库的log10函数进行，即对数函数，eg: a = log10(100) + 1;

#include <cmath>
#include <cstdio>
#include <string>
#include <cstring>
#include <iostream>
#include <algorithm>

using namespace std;

int a, b, lena, lenb;
int S[11], cnt;
bool vis;

bool isPrime(int x)
{
    if(x < 2) return false;
    for(int i = 2; i * i <= x; i++)
        if(x % i == 0) return false;
    return true;
}

//x:目前枚举位置 k:回文数位数
void dfs(int x, int k)
{
    //(k+1)/2 为回文数位数的中间值
    if(x == (k + 1)/2)
    {
        //将回文数中间值以前的数据复制到中间以后
        for(int i = k; i > x; i--) S[i] = S[k - i + 1];
        //将构造的回文数变成整数
        int sum = 0;
        for(int i = 1; i <= k; i++) sum = sum * 10 + S[i];
        //当前回文数小于a，跳过；
        if(sum < a) return;
        //当前回文数大于b，跳出(vis标记跳出，即后面不可能有符合条件的数)
        if(sum > b) {vis = false;return;}
        //当前回文数是质数输出
        if(isPrime(sum)) cout << sum << endl;
        //结束本次构造继续下一次
        return;
    }
    //x=0 说明目前构造的是首位，首位不能为0
    int pos;
    if(x == 0) pos = 1; else pos = 0;
    for(int i = pos; i <= 9; i++)
    {
        //若vis已标记为跳出则直接return
        if(vis == false) return;
        //存值(下标从1开始)
        S[x + 1] = i;
        //dfs下一位
        dfs(x + 1, k);
    }
    return;
}

int main()
{
    cin >> a >> b;
    //cmath库的log函数可用来求位数
    lena = log10(a) + 1;
    lenb = log10(b) + 1;
    //构造从lena位到lenb位，构造比a小return，比b大跳出，a~b时判断是否是质数
    for(int i = lena; i <= lenb;i++)
    {
        //偶位数的回文数(除了11)不可能是质数
        if(i != 2 && i % 2 == 0) continue;
        //i=9 只有1亿不是质数
        if(i == 9) continue;
        //从第0位开始搜i位的回文数,更新vis值
        vis = true;
        dfs(0, i);
    }
    return 0;
}