Smarandache-Wellin Sequence in C++
Introduction:
In number theory, the Smarandache-Wellin sequence (SWS) is a fascinating sequence where each term is formed by concatenating the first n prime numbers. This tutorial delves into the C++ programming language to explore this sequence, guiding us through its generation, properties, and potential applications. We'll cover efficient prime number generation, string manipulation, and considerations for handling large numbers.
Smarandache-Wellin Numbers
- The first term of the SWS is simply 2, the first prime number.
- The second term, 23, is obtained by concatenating the first two primes (2 and 3).
- Subsequent terms follow the same pattern, concatenating the first n primes (e.g., 235, 2357, ...).
Properties of the SWS
- The SWS is an infinite sequence with no known formula to determine its terms directly.
- The primality of SWS terms is a complex question; only the first few terms are known to be prime (2, 23, 2357).
Applications of Smarandache-Wellin sequence (SWS):
The SWS is primarily an object of mathematical curiosity, but it might have potential applications in areas like cryptography due to its terms' large and somewhat unpredictable nature. However, further research is needed to explore this possibility.
Example:
Let us take an example to illustrate the Smarandache-Wellin sequence (SWS) in C++.
#include <iostream>
#include <vector>
#include <string>
using namespace std;
bool isPrime(int num) {
if (num <= 1) return false;
for (int i = 2; i * i <= num; ++i) {
if (num % i == 0) return false;
}
return true;
}
vector<int> generatePrimes(int n) {
vector<int> primes;
int count = 0;
int num = 2;
while (count < n) {
if (isPrime(num)) {
primes.push_back(num);
count++;
}
num++;
}
return primes;
}
string generateSWTerm(const vector<int>& primes) {
string term;
for (int prime : primes) {
term += to_string(prime);
}
return term;
}
int main() {
int n;
cout << "Enter the number of terms: ";
cin >> n;
vector<int> primes = generatePrimes(n);
cout << "First " << n << " terms of the Smarandache-Wellin sequence:" << endl;
for (int i = 0; i < n; ++i) {
string term = generateSWTerm(vector<int>(primes.begin(), primes.begin() + i + 1));
cout << term << " ";
}
cout << endl;
return 0;
}
Output:
![Smarandache-Wellin Sequence in C++/>
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