给定两个整数A和B以及一个整数N。任务是找出形式为A + nB或B + nA的N个素数( n = 1、2、3…)。如果不可能,请打印-1。
例子:
Input: A = 3, B = 5, N = 4
Output: 13, 11, 17, 23
Explanation:
13 (3+2*5)
11 (5+2*3)
17 (5+4*3)
23 (3+4*5)
Input: A = 4, B = 6, N = 4
Output: -1
方法:
由于A + nB是质数,因此可以确定A和B之间不应存在任何公因子,这意味着A和B应该是互质的。
最好,最有效的方法是使用Dirichlet定理。
Dirichlet定理说,如果a和b是相对质数的正整数,则算术序列a , a + b , a + 2b , a + 3b …包含无限多个质数。
因此,首先检查A和B是否互质。
如果A和B是互质的,则检查A + nB和B + nA的素数,其中n = 1、2、3…。打印其中的前N个质数。
下面是上述方法的实现:
C++
// C++ implementation of
// the above approach
#include
using namespace std;
// Utility function to check
// whether two numbers is
// co-prime or not
int coprime(int a, int b)
{
if (__gcd(a, b) == 1)
return true;
else
return false;
}
// Utility function to check
// whether a number is prime
// or not
bool isPrime(int n)
{
// Corner case
if (n <= 1)
return false;
if (n == 2 or n == 3)
return true;
// Check from 2 to sqrt(n)
for (int i = 2; i * i <= n; i++)
if (n % i == 0)
return false;
return true;
}
// finding the Prime numbers
void findNumbers(int a, int b, int n)
{
bool possible = true;
// Checking whether given
// numbers are co-prime
// or not
if (!coprime(a, b))
possible = false;
int c1 = 1;
int c2 = 1;
int num1, num2;
// To store the N primes
set st;
// If 'possible' is true
if (possible) {
// Printing n numbers
// of prime
while ((int)st.size() != n) {
// checking the form of a+nb
num1 = a + (c1 * b);
if (isPrime(num1)) {
st.insert(num1);
}
c1++;
// Checking the form of b+na
num2 = b + (c2 * a);
if (isPrime(num2)) {
st.insert(num2);
}
c2++;
}
for (int i : st)
cout << i << " ";
}
// If 'possible' is false
// return -1
else
cout << "-1";
}
// Driver Code
int main()
{
int a = 3;
int b = 5;
int n = 4;
findNumbers(a, b, n);
return 0;
}
Java
// Java implementation of the approach
import java.util.*;
class GFG
{
static int __gcd(int a, int b)
{
if (b == 0)
return a;
return __gcd(b, a % b);
}
// Utility function to check
// whether two numbers is
// co-prime or not
static boolean coprime(int a, int b)
{
if (__gcd(a, b) == 1)
return true;
else
return false;
}
// Utility function to check
// whether a number is prime
// or not
static boolean isPrime(int n)
{
// Corner case
if (n <= 1)
return false;
if (n == 2 || n == 3)
return true;
// Check from 2 to sqrt(n)
for (int i = 2; i * i <= n; i++)
if (n % i == 0)
return false;
return true;
}
// finding the Prime numbers
static void findNumbers(int a, int b, int n)
{
boolean possible = true;
// Checking whether given
// numbers are co-prime
// or not
if (!coprime(a, b))
possible = false;
int c1 = 1;
int c2 = 1;
int num1, num2;
// To store the N primes
HashSet st = new HashSet();
// If 'possible' is true
if (possible)
{
// Printing n numbers
// of prime
while ((int)st.size() != n)
{
// checking the form of a+nb
num1 = a + (c1 * b);
if (isPrime(num1))
{
st.add(num1);
}
c1++;
// Checking the form of b+na
num2 = b + (c2 * a);
if (isPrime(num2))
{
st.add(num2);
}
c2++;
}
for (int i : st)
System.out.print(i + " ");
}
// If 'possible' is false
// return -1
else
System.out.print("-1");
}
// Driver Code
public static void main(String[] args)
{
int a = 3;
int b = 5;
int n = 4;
findNumbers(a, b, n);
}
}
// This code is contributed by Rajput-Ji
Python3
# Python3 implementation of the above approach
from math import gcd, sqrt
# Utility function to check
# whether two numbers is
# co-prime or not
def coprime(a, b) :
if (gcd(a, b) == 1) :
return True;
else :
return False;
# Utility function to check
# whether a number is prime
# or not
def isPrime(n) :
# Corner case
if (n <= 1) :
return False;
if (n == 2 or n == 3) :
return True;
# Check from 2 to sqrt(n)
for i in range(2, int(sqrt(n)) + 1) :
if (n % i == 0) :
return False;
return True;
# finding the Prime numbers
def findNumbers(a, b, n) :
possible = True;
# Checking whether given
# numbers are co-prime
# or not
if (not coprime(a, b)) :
possible = False;
c1 = 1;
c2 = 1;
num1 = 0;
num2 = 0;
# To store the N primes
st = set();
# If 'possible' is true
if (possible) :
# Printing n numbers
# of prime
while (len(st) != n) :
# checking the form of a+nb
num1 = a + (c1 * b);
if (isPrime(num1)):
st.add(num1);
c1 += 1;
# Checking the form of b+na
num2 = b + (c2 * a);
if (isPrime(num2)):
st.add(num2);
c2 += 1;
for i in st :
print(i, end = " ");
# If 'possible' is false
# return -1
else :
print("-1");
# Driver Code
if __name__ == "__main__" :
a = 3;
b = 5;
n = 4;
findNumbers(a, b, n);
# This code is contributed by AnkitRai01
C#
// C# implementation of the approach
using System;
using System.Collections.Generic;
class GFG
{
static int __gcd(int a, int b)
{
if (b == 0)
return a;
return __gcd(b, a % b);
}
// Utility function to check
// whether two numbers is
// co-prime or not
static bool coprime(int a, int b)
{
if (__gcd(a, b) == 1)
return true;
else
return false;
}
// Utility function to check
// whether a number is prime
// or not
static bool isPrime(int n)
{
// Corner case
if (n <= 1)
return false;
if (n == 2 || n == 3)
return true;
// Check from 2 to sqrt(n)
for (int i = 2; i * i <= n; i++)
if (n % i == 0)
return false;
return true;
}
// finding the Prime numbers
static void findNumbers(int a, int b, int n)
{
bool possible = true;
// Checking whether given
// numbers are co-prime
// or not
if (!coprime(a, b))
possible = false;
int c1 = 1;
int c2 = 1;
int num1, num2;
// To store the N primes
HashSet st = new HashSet();
// If 'possible' is true
if (possible)
{
// Printing n numbers
// of prime
while (st.Count != n)
{
// checking the form of a+nb
num1 = a + (c1 * b);
if (isPrime(num1))
{
st.Add(num1);
}
c1++;
// Checking the form of b+na
num2 = b + (c2 * a);
if (isPrime(num2))
{
st.Add(num2);
}
c2++;
}
foreach (int i in st)
Console.Write(i + " ");
}
// If 'possible' is false
// return -1
else
Console.Write("-1");
}
// Driver Code
public static void Main(String[] args)
{
int a = 3;
int b = 5;
int n = 4;
findNumbers(a, b, n);
}
}
// This code is contributed by 29AjayKumar
输出:
11 13 17 23
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