大于 1 的整数 p 是素数，当当 p 的唯一除数是 1 和 p。前几个素数是 2, 3, 5, 7, 11, 13, …
Lehmann 检验是对自然数 n 的概率素性检验，它可以检验任何一种数的素性(无论大奇数是否为素数)。莱曼检验是费马素性检验的变体。
使用的方法如下：
如果 ‘n’ 是一个奇数并且 ‘a’ 是一个小于 n 但大于 1 的随机整数，那么
x = (a^((n-1)/2)) (mod n)它是计算出来的。
- 如果 x 是 1 或 -1(或 n-1)，则 n 可能是素数。
- 如果 x 不是 1 或 -1(或 n-1)，则 n 肯定是合数。
在这种情况下，任何合数都可以被证明是素数这一事实取决于随机值“a”。如果 a 和 n 的所有值互质，则 n 可以说是质数。
Example-1:
Input: n = 13
Output: 13 is Prime
Explanation:
Let a = 3, then,
3^((13-1)/2) % 13 = 729 % 13 = 1
Hence, 13 is Prime.
Example-2:
Input: n = 91
Output: 91 is Composite
Explanation:
Let a = 3, then,
3^((91-1)/2) % 91 = 27
Hence, 91 is Composite. C++
// C++ code for Lehmann's Primality Test
#include
#include
#include
#include
using namespace std;
// function to check Lehmann's test
int lehmann(int n, int t)
{
// generating a random base less than n
int a = 2 + (rand() % (n - 1));
// calculating exponent
int e = (n - 1) / 2;
// iterate to check for different base values
// for given number of tries 't'
while(t > 0)
{
// calculating final value using formula
int result =((int)(pow(a, e)))% n;
//if not equal, try for different base
if((result % n) == 1 || (result % n) == (n - 1))
{
a = 2 + (rand() % (n - 1));
t -= 1;
}
// else return negative
else
return -1;
}
// return positive after attempting
return 1;
}
// Driver code
int main()
{
int n = 13 ; // number to be tested
int t = 10 ; // number of tries
// if n is 2, it is prime
if(n == 2)
cout << "2 is Prime.";
// if even, it is composite
if(n % 2 == 0)
cout << n << " is Composite";
// if odd, check
else
{
int flag = lehmann(n, t);
if(flag ==1)
cout << n << " may be Prime.";
else
cout << n << " is Composite.";
}
}
// This code is contibuted by chitranayal Java
// Java code for Lehmann's Primality Test
// importing "random" for random operations
import java.util.Random;
class GFG
{
// function to check Lehmann's test
static int lehmann(int n, int t)
{
// create instance of Random class
Random rand = new Random();
// generating a random base less than n
int a = rand.nextInt(n - 3) + 2;
// calculating exponent
float e = (n - 1) / 2;
// iterate to check for different base values
// for given number of tries 't'
while(t > 0)
{
// calculating final value using formula
int result = ((int)(Math.pow(a, e))) % n;
// if not equal, try for different base
if((result % n) == 1 || (result % n) == (n - 1))
{
a = rand.nextInt(n - 3) + 2;
t -= 1;
}
// else return negative
else
return -1;
}
// return positive after attempting
return 1;
}
// Driver code
public static void main (String[] args)
{
int n = 13; // number to be tested
int t = 10; // number of tries
// if n is 2, it is prime
if(n == 2)
System.out.println(" 2 is Prime.");
// if even, it is composite
if(n % 2 == 0)
System.out.println(n + " is Composite");
// if odd, check
else
{
long flag = lehmann(n, t);
if(flag == 1)
System.out.println(n + " may be Prime.");
else
System.out.println(n + " is Composite.");
}
}
}
// This code is contributed by AnkitRai01Python3
# Python code for Lehmann's Primality Test
# importing "random" for random operations
import random
# function to check Lehmann's test
def lehmann(n, t):
# generating a random base less than n
a = random.randint(2, n-1)
# calculating exponent
e =(n-1)/2
# iterate to check for different base values
# for given number of tries 't'
while(t>0):
# calculating final value using formula
result =((int)(a**e))% n
# if not equal, try for different base
if((result % n)== 1 or (result % n)==(n-1)):
a = random.randint(2, n-1)
t-= 1
# else return negative
else:
return -1
# return positive after attempting
return 1
# Driver code
n = 13 # number to be tested
t = 10 # number of tries
# if n is 2, it is prime
if(n is 2):
print("2 is Prime.")
# if even, it is composite
if(n % 2 == 0):
print(n, "is Composite")
# if odd, check
else:
flag = lehmann(n, t)
if(flag is 1):
print(n, "may be Prime.")
else:
print(n, "is Composite.")C#
// C# code for Lehmann's Primality Test
using System;
class GFG
{
// function to check Lehmann's test
static int lehmann(int n, int t)
{
// create instance of Random class
Random rand = new Random();
// generating a random base less than n
int a = rand.Next(n - 3) + 2;
// calculating exponent
float e = (n - 1) / 2;
// iterate to check for different base values
// for given number of tries 't'
while(t > 0)
{
// calculating final value using formula
int result = ((int)(Math.Pow(a, e))) % n;
// if not equal, try for different base
if((result % n) == 1 ||
(result % n) == (n - 1))
{
a = rand.Next(n - 3) + 2;
t -= 1;
}
// else return negative
else
return -1;
}
// return positive after attempting
return 1;
}
// Driver code
public static void Main (String[] args)
{
int n = 13; // number to be tested
int t = 10; // number of tries
// if n is 2, it is prime
if(n == 2)
Console.WriteLine(" 2 is Prime.");
// if even, it is composite
if(n % 2 == 0)
Console.WriteLine(n + " is Composite");
// if odd, check
else
{
long flag = lehmann(n, t);
if(flag == 1)
Console.WriteLine(n + " may be Prime.");
else
Console.WriteLine(n + " is Composite.");
}
}
}
// This code is contributed by Rajput-Ji输出：
13 may be Prime.