C＃，10039 28164位
6 0{28157} 169669
编辑：我已经基于Qualtagh的算法做了另一个程序，做了一些小的修改：
- 我正在搜索L000 ... 000R形式的数字，其中L是左组合，R是右组合。我允许左复合数字L包含多个数字，尽管这主要是一种风格上的变化，并且可能不会影响算法的效率。
- 我添加了多线程来加快搜索速度。
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
using System.Threading;
using System.Threading.Tasks;
using Mpir.NET;
class Program
{
const int PrimeNotFound = int.MaxValue;
private static BitArray _primeSieve;
private static HashSet<Tuple<int, int>> _templatesToSkip = new HashSet<Tuple<int, int>>();
static void Main(string[] args)
{
int bestDigitCount = 0;
foreach (Tuple<int, int> template in GetTemplates())
{
int left = template.Item1;
int right = template.Item2;
if (SkipTemplate(left, right))
continue;
int zeroCount = GetZeroCountOfPrime(left, right);
if (zeroCount != PrimeNotFound)
{
int digitCount = left.ToString().Length + right.ToString().Length + zeroCount;
if (digitCount >= bestDigitCount)
{
string primeStr = left + " 0{" + zeroCount + "} " + right;
Console.WriteLine("testing " + primeStr);
bool isFragile = IsFragile(left, right, zeroCount);
Console.WriteLine(primeStr + " is fragile: " + isFragile);
if (isFragile)
bestDigitCount = digitCount;
}
_templatesToSkip.Add(template);
}
}
}
private static int GetZeroCountOfPrime(int left, int right)
{
_zeroCount = 0;
int threadCount = Environment.ProcessorCount;
Task<int>[] tasks = new Task<int>[threadCount];
for (int i = 0; i < threadCount; i++)
tasks[i] = Task.Run(() => InternalGetZeroCountOfPrime(left, right));
Task.WaitAll(tasks);
return tasks.Min(task => task.Result);
}
private static int _zeroCount;
private static int InternalGetZeroCountOfPrime(int left, int right)
{
const int maxZeroCount = 40000;
int zeroCount = Interlocked.Increment(ref _zeroCount);
while (zeroCount <= maxZeroCount)
{
if (zeroCount % 1000 == 0)
Console.WriteLine("testing " + left + " 0{" + zeroCount + "} " + right);
if (IsPrime(left, right, zeroCount))
{
Interlocked.Add(ref _zeroCount, maxZeroCount);
return zeroCount;
}
else
zeroCount = Interlocked.Increment(ref _zeroCount);
}
return PrimeNotFound;
}
private static bool SkipTemplate(int left, int right)
{
for (int leftDiv = 1; leftDiv <= left; leftDiv *= 10)
for (int rightDiv = 1; rightDiv <= right; rightDiv *= 10)
if (_templatesToSkip.Contains(Tuple.Create(left / leftDiv, right % (rightDiv * 10))))
return true;
return false;
}
private static bool IsPrime(int left, int right, int zeroCount)
{
return IsPrime(left.ToString() + new string('0', zeroCount) + right.ToString());
}
private static bool IsPrime(string left, string right, int zeroCount)
{
return IsPrime(left + new string('0', zeroCount) + right);
}
private static bool IsPrime(string s)
{
using (mpz_t n = new mpz_t(s))
{
return n.IsProbablyPrimeRabinMiller(20);
}
}
private static bool IsFragile(int left, int right, int zeroCount)
{
string leftStr = left.ToString();
string rightStr = right.ToString();
for (int startIndex = 0; startIndex < leftStr.Length - 1; startIndex++)
for (int count = 1; count < leftStr.Length - startIndex; count++)
if (IsPrime(leftStr.Remove(startIndex, count), rightStr, zeroCount))
return false;
for (int startIndex = 1; startIndex < rightStr.Length; startIndex++)
for (int count = 1; count <= rightStr.Length - startIndex; count++)
if (IsPrime(leftStr, rightStr.Remove(startIndex, count), zeroCount))
return false;
return true;
}
private static IEnumerable<Tuple<int, int>> GetTemplates()
{
const int maxDigitCount = 8;
PreparePrimeSieve((int)BigInteger.Pow(10, maxDigitCount));
for (int digitCount = 2; digitCount <= maxDigitCount; digitCount++)
{
for (int leftCount = 1; leftCount < digitCount; leftCount++)
{
int rightCount = digitCount - leftCount;
int maxLeft = (int)BigInteger.Pow(10, leftCount);
int maxRight = (int)BigInteger.Pow(10, rightCount);
for (int left = maxLeft / 10; left < maxLeft; left++)
for (int right = maxRight / 10; right < maxRight; right++)
if (IsValidTemplate(left, right, leftCount, rightCount))
yield return Tuple.Create(left, right);
}
}
}
private static void PreparePrimeSieve(int limit)
{
_primeSieve = new BitArray(limit + 1, true);
_primeSieve[0] = false;
_primeSieve[1] = false;
for (int i = 2; i * i <= limit; i++)
if (_primeSieve[i])
for (int j = i * i; j <= limit; j += i)
_primeSieve[j] = false;
}
private static bool IsValidTemplate(int left, int right, int leftCount, int rightCount)
{
int rightDigit = right % 10;
if ((rightDigit != 1) && (rightDigit != 9))
return false;
if (left % 10 == 0)
return false;
if ((left + right) % 3 == 0)
return false;
if (!Coprime(left, right))
return false;
int leftDiv = 1;
for (int i = 0; i <= leftCount; i++)
{
int rightDiv = 1;
for (int j = 0; j <= rightCount; j++)
{
int combination = left / leftDiv * rightDiv + right % rightDiv;
if (_primeSieve[combination])
return false;
rightDiv *= 10;
}
leftDiv *= 10;
}
return true;
}
private static bool Coprime(int a, int b)
{
while (b != 0)
{
int t = b;
b = a % b;
a = t;
}
return a == 1;
}
}
旧答案：
8 0{5436} 4 0{4600} 1
这是一些易碎素数的著名模式：
600..00X00..009
900..00X00..009
800..00X00..001
999..99X99..999
其中X可以是1、2、4、5、7或8。
对于此类数字，我们仅需考虑（长度-1）种可能的Remove操作。另一个Remove运算会产生重复数或明显是合成数。我尝试搜索最多800个数字的所有此类数字，并发现出现了4种模式，而其余频率则更高：8007001、8004001、9997999和6004009。由于Emil和Jakube使用的是999X999模式，因此我决定只使用8004001添加一些变化。
我在算法中添加了以下优化：
- 我从具有7000位数字的数字开始搜索，然后每次发现易碎的质数时将长度增加1500。如果没有给定长度的易碎素数，则将其递增1。7000和1500只是看起来合适的任意数字。
- 我正在使用多线程来同时搜索具有不同长度的数字。
- 每个主要检查的结果都存储在哈希表中，以防止重复检查。
- 我正在使用Mpir.NET的 Miller-Rabin实现，该实现非常快（MPIR是GMP的分支）。
using System;
using System.Collections.Concurrent;
using System.Threading.Tasks;
using Mpir.NET;
class Program
{
const string _template = "8041";
private static ConcurrentDictionary<Tuple<int, int>, byte> _compositeNumbers = new ConcurrentDictionary<Tuple<int, int>, byte>();
private static ConcurrentDictionary<int, int> _leftPrimes = new ConcurrentDictionary<int, int>();
private static ConcurrentDictionary<int, int> _rightPrimes = new ConcurrentDictionary<int, int>();
static void Main(string[] args)
{
int threadCount = Environment.ProcessorCount;
Task[] tasks = new Task[threadCount];
for (int i = 0; i < threadCount; i++)
{
int index = i;
tasks[index] = Task.Run(() => SearchFragilePrimes());
}
Task.WaitAll(tasks);
}
private const int _lengthIncrement = 1500;
private static int _length = 7000;
private static object _lengthLock = new object();
private static object _consoleLock = new object();
private static void SearchFragilePrimes()
{
int length;
lock (_lengthLock)
{
_length++;
length = _length;
}
while (true)
{
lock (_consoleLock)
{
Console.WriteLine("{0:T}: length = {1}", DateTime.Now, length);
}
bool found = false;
for (int rightCount = 1; rightCount <= length - 2; rightCount++)
{
int leftCount = length - rightCount - 1;
if (IsFragilePrime(leftCount, rightCount))
{
lock (_consoleLock)
{
Console.WriteLine("{0:T}: {1} {2}{{{3}}} {4} {2}{{{5}}} {6}",
DateTime.Now, _template[0], _template[1], leftCount - 1,
_template[2], rightCount - 1, _template[3]);
}
found = true;
break;
}
}
lock (_lengthLock)
{
if (found && (_length < length + _lengthIncrement / 2))
_length += _lengthIncrement;
else
_length++;
length = _length;
}
}
}
private static bool IsFragilePrime(int leftCount, int rightCount)
{
int count;
if (_leftPrimes.TryGetValue(leftCount, out count))
if (count < rightCount)
return false;
if (_rightPrimes.TryGetValue(rightCount, out count))
if (count < leftCount)
return false;
if (!IsPrime(leftCount, rightCount))
return false;
for (int i = 0; i < leftCount; i++)
if (IsPrime(i, rightCount))
return false;
for (int i = 0; i < rightCount; i++)
if (IsPrime(leftCount, i))
return false;
return true;
}
private static bool IsPrime(int leftCount, int rightCount)
{
Tuple<int, int> tuple = Tuple.Create(leftCount, rightCount);
if (_compositeNumbers.ContainsKey(tuple))
return false;
using (mpz_t n = new mpz_t(BuildStr(leftCount, rightCount)))
{
bool result = n.IsProbablyPrimeRabinMiller(20);
if (result)
{
_leftPrimes.TryAdd(leftCount, rightCount);
_rightPrimes.TryAdd(rightCount, leftCount);
}
else
_compositeNumbers.TryAdd(tuple, 0);
return result;
}
}
private static string BuildStr(int leftCount, int rightCount)
{
char[] chars = new char[leftCount + rightCount + 1];
for (int i = 0; i < chars.Length; i++)
chars[i] = _template[1];
chars[0] = _template[0];
chars[leftCount + rightCount] = _template[3];
chars[leftCount] = _template[2];
return new string(chars);
}
}