Kynea Numbers In C++

Kynea numbers are a special class of numbers in mathematics, defined as numbers of the form:

Kn=(2n+1)2−2

where n is a non-negative integer. These numbers have unique properties and are part of the study of number theory.

Understanding Kynea Numbers

To understand Kynea numbers better, let’s break down their mathematical expression:

• ✅ Base Expression: Start with 2n, which involves exponentiation.
• ✅ Increment: Add 1 to 2n, yielding 2n +1.
• ✅ Square the Result: Square the sum to produce (2n+1)2
• ✅ Subtract 2: Finally, subtract 2 to compute the Kynea number.

For example:

• ✅ For n = 0: K0 = (2 0 + 1) 2 −2 = (1 + 1) 2 −2 = 4 − 2 = 2.
• ✅ For n = 1: K1 = (2 1 + 1) 2 −2 = (2 + 1) 2 −2 = 9 − 2 = 7.
• ✅ For n = 2: K2 = (2 2 + 1) 2 −2 = (4 + 1) 2 −2 = 25 − 2 = 23.

Key Characteristics of Kynea Numbers:

• ✅ Growth: Kynea numbers snowball with increasing n due to the squaring operation•
• ✅ Oddness: All Kynea numbers are odd because the term (2n+1)2 is always odd, and subtracting 2 from a bizarre number retains oddness•
• ✅ Applications: Kynea numbers are of interest in primality testing, sequence generation, and cryptographic studies•

Example:

Let us take an example to illustrate the Kynea Number in C++ .

#include <iostream>
#include <cmath> // For pow function
#include <vector>

using namespace std;

// Function to compute Kynea number for a given n
long long computeKyneaNumber(int n) {
    long long base = pow(2, n) + 1; // Calculate 2^n + 1
    return (base * base) - 2;       // Square it and subtract 2
}

// Function to generate a list of Kynea numbers up to a given limit
vector<long long> generateKyneaNumbers(int limit) {
    vector<long long> kyneaNumbers;
    for (int i = 0; i <= limit; ++i) {
        kyneaNumbers.push_back(computeKyneaNumber(i));
    }
    return kyneaNumbers;
}

int main() {
    int limit = 5; // Directly set the limit for n

    // Generate and display Kynea numbers
    vector<long long> kyneaNumbers = generateKyneaNumbers(limit);
    cout << "Kynea numbers up to n = " << limit << " are:" << endl;
    for (int i = 0; i < kyneaNumbers.size(); ++i) {
        cout << "K_" << i << " = " << kyneaNumbers[i] << endl;
    }

    return 0;
}

Output:

Kynea numbers up to n = 5 are:
K_0 = 2
K_1 = 7
K_2 = 23
K_3 = 79
K_4 = 287
K_5 = 1087

Algorithm:

Understand the Formula. The Kynea number formula is:

Kn=(2 n +1) 2 −2

This formula calculates a Kynea number for a given.

2 n : Raise 2 to the power of

+1: Add 1 to the result of 2 n

2(Square): Square the value from the previous step•

-2: Subtract 2 to get the final Kynea number•

Fix the Range of Computation: Instead of waiting for user input, we directly decide how many Kynea numbers to calculate• For example, if the range is from n 0 to n = 5, the program will compute the first 6 Kynea numbers.

Compute Kynea Numbers for Each n

For every n in the range:

• ✅ Raise 2 to the power n : Start by calculating 2 n . This snowballs as n increases.
• ✅ Add 1 : Add 1 to the result of 2 n, giving 2 n + 1.
• ✅ Square the Value : Multiply (2 n + 1) by itself to get (2 n + 1)
• ✅ Subtract 2 : Finally, subtract 2 to calculate K n = (2 n + 1) 2 - 2.

Store Results: Save the computed Kynea numbers in a collection (like a list or vector )• This helps keep track of all the numbers and makes them easier to display later•

Display the Results: For each Kynea number, print its position n and the corresponding value Kn. This shows the relationship between n and the rapidly growing Kynea numbers.

Complexity Analysis:

Time Complexity:

The time complexity of computing Kynea numbers is O(n) for generating n numbers. For each number, calculating 2n using the pow functions is O(log n). Since this is repeated n times, the overall complexity becomes O(n. log n).

Space Complexity:

The space complexity of the Kynea number algorithm is O(n) due to the storage of the Kynea numbers in a vector• Each number is stored in memory, so as the number of computations grows, the space required increases linearly with the size of the input limit n•