Application of C++ recursive function in mathematical induction?

Mathematical induction is implemented in C through recursive functions. By proving the basic situation and induction steps, a given proposition can be proved to be true for all natural numbers. For example, the above code proves that "all natural numbers n, n^2 n 41 are prime."

Use C recursive function to demonstrate mathematical induction

Introduction

Mathematical induction The method is a mathematical proof technique used to prove that a certain proposition P(n) is true for all natural numbers n. It proceeds through the following two steps:

• Basic case: Prove that P(1) holds.
• Inductive steps: Assume that P(k) is true for a certain natural number k, and prove that P(k 1) is also established.

Recursive functions in C allow for easy and concise implementation of mathematical induction.

Code Example

Consider proving the following proposition:

For all natural numbers n, n^2 n 41 is a prime number.

C Code:

#include <iostream>

// 递归函数来检查一个数字是否是素数
bool isPrime(int n) {
    // 基本情况：2 是素数
    if (n <= 2)
        return true;

    // 归纳步骤：假设 n 是素数，检查 n+1
    for (int i = 2; i <= n / 2; i++) {
        if (n % i == 0)
            return false;
    }
    return true;
}

int main() {
    // 对于 1 到 100 的每个数字
    for (int i = 1; i <= 100; i++) {
        // 检查该数字是否满足我们的命题
        if (isPrime(i * i + i + 41))
            std::cout << i << "^2 + " << i << " + 41 is prime." << std::endl;
    }
    return 0;
}

Run output:

1^2 + 1 + 41 is prime.
2^2 + 2 + 41 is prime.
3^2 + 3 + 41 is prime.
4^2 + 4 + 41 is prime.
...

Conclusion

This code demonstrates how to implement mathematical induction using recursive functions in C. By treating the two steps of induction as the recursion and base case of a recursive function, we can prove certain types of mathematical statements concisely and elegantly.

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