洛谷 P1217
因为 既是一个质数又是一个回文数（从左到右和从右到左是看一样的），所以 是回文质数。
写一个程序来找出范围 （一亿）间的所有回文质数。
第一行输入两个正整数 和 。
输出一个回文质数的列表，一行一个。
5 500
5 7 11 101 131 151 181 191 313 353 373 383
Hint 1: Generate the palindromes and see if they are prime.
提示 1: 找出所有的回文数再判断它们是不是质数（素数）.
Hint 2: Generate palindromes by combining digits properly. You might need more than one of the loops like below.
提示 2: 要产生正确的回文数，你可能需要几个像下面这样的循环。
题目翻译来自NOCOW。
USACO Training Section 1.5
产生长度为 的回文数：
cppfor (d1 = 1; d1 <= 9; d1+=2) { // 只有奇数才会是素数
for (d2 = 0; d2 <= 9; d2++) {
for (d3 = 0; d3 <= 9; d3++) {
palindrome = 10000*d1 + 1000*d2 +100*d3 + 10*d2 + d1;//(处理回文数...)
}
}
}
看这个数字范围，如果直接遍历然后判断素数和回文数会很慢，可以考虑构造回文数然后提素数。
因此，对这个过程加以优化，可以简单写出以下代码
cpp#include <iostream>
#include <algorithm>
#include <cmath>
bool is_prime_ge5(int n) {
int sq = floor(sqrt(n));
for (int num = sq | 1; num > 1; num -= 2) {
if (n % num == 0) {
return false;
}
}
return true;
}
int main() {
#ifdef OHTOAI_LOCAL_MACHINE
std::freopen("input.txtendlr", stdin);
std::freopen("output.txtendlw", stdout);
#endif
using std::cin;
using std::cout;
using std::endl;
std::ios::sync_with_stdio(false);
int a, b;
int palindrome_withen_100[] = { 5, 7, 11 };
cin >> a >> b;
for (auto n : palindrome_withen_100) {
if (a <= n && n <= b)
cout << n << endl;
}
for (int dt1 = 1; dt1 <= 9; dt1 += 2) {
int num1 = dt1 * 101;
if (num1 + 101 < a)
continue;
for (int dt2 = 0; dt2 <= 9; dt2++) {
int num = num1 + dt2 * 10;
if (num + 10 < a)
continue;
else if (b < num)
break;
else if (is_prime_ge5(num))
cout << num << endl;
}
}
for (int dt1 = 1; dt1 <= 9; dt1 += 2) {
int num1 = dt1 * 10001;
if (num1 + 10001 < a)
continue;
for (int dt2 = 0; dt2 <= 9; dt2++) {
int num2 = num1 + dt2 * 1010;
if (num2 + 1010 < a)
continue;
for (int dt3 = 0; dt3 <= 9; dt3++) {
int num = num2 + dt3 * 100;
if (num + 100 < a)
continue;
else if (b < num)
break;
else if (is_prime_ge5(num)) {
cout << num << endl;
}
}
}
}
for (int dt1 = 1; dt1 <= 9; dt1 += 2) {
int num1 = dt1 * 1000001;
if (num1 + 1000001 < a)
continue;
for (int dt2 = 0; dt2 <= 9; dt2++) {
int num2 = num1 + dt2 * 100010;
if (num2 + 100010 < a)
continue;
for (int dt3 = 0; dt3 <= 9; dt3++) {
int num3 = num2 + dt3 * 10100;
if (num3 + 10100 < a)
continue;
for (int dt4 = 0; dt4 <= 9; dt4++) {
int num = num3 + dt4 * 1000;
if (num + 1000 < a)
continue;
else if (b < num)
break;
else if (is_prime_ge5(num)) {
cout << num << endl;
}
}
}
}
}
return 0;
}
容易得到1亿范围内的所有回文质数，因此很容易写出以下的代码
cpp#include <algorithm>
#include <cstdio>
int main() {
constexpr int list[] = {
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,
};
constexpr auto Length = sizeof(list) / sizeof(list[0]);
int a, b;
scanf("%d %d", &a, &b);
auto low = std::lower_bound(list, list + Length, a);
auto high = std::upper_bound(low, list + Length, b);
for (auto e = low; e != high; ++e) {
printf("%d\n", *e);
}
return 0;
}
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