可截断素数（Truncatable primes）

可截断素数是当从一端连续删除数字时，得到的仍旧是素数的一个素数。

举例

997是左截短素数，因为997, 97 和 7 都是素数。
7393是右截断素数，因为7393, 739, 73 和 7 都是素数。
在可截断质数中不允许有零。

任务

找出小于100万的最大的左可截断和右可截断素数。

相关文章：
寻找给定基数中的最大左可截断素数

C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>#define MAX_PRIME 1000000
char *primes;
int n_primes;/*  Sieve. If we were to handle 10^9 range, use bit field. Regardless,*  if a large amount of prime numbers need to be tested, sieve is fast.*/
void init_primes()
{int j;primes = malloc(sizeof(char) * MAX_PRIME);memset(primes, 1, MAX_PRIME);primes[0] = primes[1] = 0;int i = 2;while (i * i < MAX_PRIME) {for (j = i * 2; j < MAX_PRIME; j += i)primes[j] = 0;while (++i < MAX_PRIME && !primes[i]);}
}int left_trunc(int n)
{int tens = 1;while (tens < n) tens *= 10;while (n) {if (!primes[n]) return 0;tens /= 10;if (n < tens) return 0;n %= tens;}return 1;
}int right_trunc(int n)
{while (n) {if (!primes[n]) return 0;n /= 10;}return 1;
}int main()
{int n;int max_left = 0, max_right = 0;init_primes();for (n = MAX_PRIME - 1; !max_left;  n -= 2)if (left_trunc(n)) max_left = n;for (n = MAX_PRIME - 1; !max_right; n -= 2)if (right_trunc(n)) max_right = n;printf("Left: %d; right: %d\n", max_left, max_right);return 0;
}

输出

Left: 998443; right: 739399

对小数(1000000不大)进行质数检验的更快的方法，并自底向上生成可处理的质数：

#include <stdio.h>#define MAXN 1000000
int maxl, maxr;int is_prime(int n)
{int p;if (n % 3 == 0) return 0;for (p = 6; p * p <= n; p += 6)if (!(n % (p + 1) && n % (p + 5)))return 0;return 1;
}void left(int n, int tens)
{int i, nn;if (n > maxl) maxl = n;if (n < MAXN / 10)for (tens *= 10, i = 1; i < 10; i++)if (is_prime(nn = i * tens + n))left(nn, tens);
}void right(int n)
{int i, nn;static int d[] = {1,3,7,9};if (n > maxr) maxr = n;if (n < MAXN / 10)for (i = 1; i < 4; i++)if (is_prime(nn = n * 10 + d[i])) right(nn);
}int main(void)
{left(3, 1); left(7, 1);right(3); right(5); right(7);printf("%d %d\n", maxl, maxr);return 0;
}

输出：

998443 739399

C#

using System;  // 4790@3.6
using System.Collections.Generic;
class truncatable_primes
{static void Main(){uint m = 1000000;Console.Write("L " + L(m) + " R " + R(m) + "  ");var sw = System.Diagnostics.Stopwatch.StartNew();for (int i = 1000; i > 0; i--) { L(m); R(m); }Console.Write(sw.Elapsed); Console.Read();}static uint L(uint n){n -= n & 1; n--;for (uint d, d1 = 100; ; n -= 2){while (n % 3 == 0 || n % 5 == 0 || n % 7 == 0) n -= 2;if ((d = n % 10) == 3 || d == 7){while (d1 < n && d < (d = n % d1) && isP(d)) d1 *= 10;if (d1 > n && isP(n)) return n; d1 = 100;}}}static uint R(uint m){var p = new List<uint>() { 2, 3, 5, 7 }; uint n = 20, np;for (int i = 1; i < p.Count; n = 10 * p[i++]){if ((np = n + 1) >= m) break; if (isP(np)) p.Add(np);if ((np = n + 3) >= m) break; if (isP(np)) p.Add(np);if ((np = n + 7) >= m) break; if (isP(np)) p.Add(np);if ((np = n + 9) >= m) break; if (isP(np)) p.Add(np);}return p[p.Count - 1];}static bool isP(uint n){if (n < 7) return n == 2 || n == 3 || n == 5;if ((n & 1) == 0 || n % 3 == 0 || n % 5 == 0) return false;for (uint r = (uint)Math.Sqrt(n), d = 7; d <= r; d += 30)if (n % (d + 00) == 0 || n % (d + 04) == 0 ||n % (d + 06) == 0 || n % (d + 10) == 0 ||n % (d + 12) == 0 || n % (d + 16) == 0 ||n % (d + 22) == 0 || n % (d + 24) == 0) return false;return true;}
}

输出：

L 998443 R 739399   24 μs

C++

#include <iostream>
#include "prime_sieve.hpp"bool is_left_truncatable(const prime_sieve& sieve, int p) {for (int n = 10, q = p; p > n; n *= 10) {if (!sieve.is_prime(p % n) || q == p % n)return false;q = p % n;}return true;
}bool is_right_truncatable(const prime_sieve& sieve, int p) {for (int q = p/10; q > 0; q /= 10) {if (!sieve.is_prime(q))return false;}return true;
}int main() {const int limit = 1000000;// find the prime numbers up to the limitprime_sieve sieve(limit + 1);int largest_left = 0;int largest_right = 0;// find largest left truncatable primefor (int p = limit; p >= 2; --p) {if (sieve.is_prime(p) && is_left_truncatable(sieve, p)) {largest_left = p;break;}}// find largest right truncatable primefor (int p = limit; p >= 2; --p) {if (sieve.is_prime(p) && is_right_truncatable(sieve, p)) {largest_right = p;break;}}// write results to standard outputstd::cout << "Largest left truncatable prime is " << largest_left << '\n';std::cout << "Largest right truncatable prime is " << largest_right << '\n';return 0;
}

prime_sieve.hpp

#ifndef PRIME_SIEVE_HPP
#define PRIME_SIEVE_HPP#include <algorithm>
#include <vector>/*** A simple implementation of the Sieve of Eratosthenes.* See https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes.*/
class prime_sieve {public:explicit prime_sieve(size_t);bool is_prime(size_t) const;
private:std::vector<bool> is_prime_;
};/*** Constructs a sieve with the given limit.** @param limit the maximum integer that can be tested for primality*/
inline prime_sieve::prime_sieve(size_t limit) {limit = std::max(size_t(3), limit);is_prime_.resize(limit/2, true);for (size_t p = 3; p * p <= limit; p += 2) {if (is_prime_[p/2 - 1]) {size_t inc = 2 * p;for (size_t q = p * p; q <= limit; q += inc)is_prime_[q/2 - 1] = false;}}
}/*** Returns true if the given integer is a prime number. The integer* must be less than or equal to the limit passed to the constructor.** @param n an integer less than or equal to the limit passed to the* constructor* @return true if the integer is prime*/
inline bool prime_sieve::is_prime(size_t n) const {if (n == 2)return true;if (n < 2 || n % 2 == 0)return false;return is_prime_.at(n/2 - 1);
}#endif

输出：

Largest left truncatable prime is 998443
Largest right truncatable prime is 739399

Go

package mainimport "fmt"func main() {sieve(1e6)if !search(6, 1e6, "left", func(n, pot int) int { return n % pot }) {panic("997?")}if !search(6, 1e6, "right", func(n, _ int) int { return n / 10 }) {panic("7393?")}
}var c []boolfunc sieve(ss int) {c = make([]bool, ss)c[1] = truefor p := 2; ; {p2 := p * pif p2 >= ss {break}for i := p2; i < ss; i += p {c[i] = true}for {p++if !c[p] {break}}}
}func search(digits, pot int, s string, truncFunc func(n, pot int) int) bool {n := pot - 1pot /= 10
smaller:for ; n >= pot; n -= 2 {for tn, tp := n, pot; tp > 0; tp /= 10 {if tn < tp || c[tn] {continue smaller}tn = truncFunc(tn, tp)}fmt.Println("max", s, "truncatable:", n)return true}if digits > 1 {return search(digits-1, pot, s, truncFunc)}return false
}

输出：

max left truncatable: 998443
max right truncatable: 739399

Java

import java.util.BitSet;public class Main {public static void main(String[] args){final int MAX = 1000000;//Sieve of Eratosthenes (using BitSet only for odd numbers)BitSet primeList = new BitSet(MAX>>1); primeList.set(0,primeList.size(),true); int sqroot = (int) Math.sqrt(MAX); primeList.clear(0); for(int num = 3; num <= sqroot; num+=2) { if( primeList.get(num >> 1) ) { int inc = num << 1;for(int factor = num * num; factor < MAX; factor += inc) { //if( ((factor) & 1) == 1) //{ primeList.clear(factor >> 1); //} } } }//Sieve ends...//Find Largest Truncatable Prime. (so we start from 1000000 - 1int rightTrunc = -1, leftTrunc = -1;for(int prime = (MAX - 1) | 1; prime >= 3; prime -= 2){if(primeList.get(prime>>1)){//Already found Right Truncatable Prime?if(rightTrunc == -1){int right = prime;while(right > 0 && right % 2 != 0 && primeList.get(right >> 1)) right /= 10;if(right == 0) rightTrunc = prime;}//Already found Left Truncatable Prime?if(leftTrunc == -1 ){//Left TruncationString left = Integer.toString(prime);if(!left.contains("0")){while( left.length() > 0 ){int iLeft = Integer.parseInt(left);if(!primeList.get( iLeft >> 1)) break;left = left.substring(1);}if(left.length() == 0) leftTrunc = prime;}}if(leftTrunc != -1 && rightTrunc != -1) //Found both? then Stop loop{break;}}}System.out.println("Left  Truncatable : " + leftTrunc);System.out.println("Right Truncatable : " + rightTrunc);}
}

输出：

Left  Truncatable : 998443
Right Truncatable : 739399

Python

maxprime = 1000000def primes(n):multiples = set()prime = []for i in range(2, n+1):if i not in multiples:prime.append(i)multiples.update(set(range(i*i, n+1, i)))return primedef truncatableprime(n):'Return a longest left and right truncatable primes below n'primelist = [str(x) for x in primes(n)[::-1]]primeset = set(primelist)for n in primelist:# n = 'abc'; [n[i:] for i in range(len(n))] -> ['abc', 'bc', 'c']alltruncs = set(n[i:] for i in range(len(n)))if alltruncs.issubset(primeset):truncateleft = int(n)breakfor n in primelist:# n = 'abc'; [n[:i+1] for i in range(len(n))] -> ['a', 'ab', 'abc']alltruncs = set([n[:i+1] for i in range(len(n))])if alltruncs.issubset(primeset):truncateright = int(n)breakreturn truncateleft, truncaterightprint(truncatableprime(maxprime))

输出：

(998443, 739399)

Ruby

def left_truncatable?(n)truncatable?(n) {|i| i.to_s[1..-1].to_i}
enddef right_truncatable?(n)truncatable?(n) {|i| i/10}
enddef truncatable?(n, &trunc_func)return false if n.to_s.include? "0"loop don = trunc_func.call(n)return true if n.zero?return false unless Prime.prime?(n)end
endrequire 'prime'
primes = Prime.each(1_000_000).to_a.reversep primes.detect {|p| left_truncatable? p}
p primes.detect {|p| right_truncatable? p}

输出：

998443
739399

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可截断素数（Truncatable primes）

C

C#

C++

Go

Java

Python

Ruby

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