Twin Prime Index Density Sequence Analyzer 2.0

1. Purpose

The primary purpose of this script is to provide a computational tool for a rigorous empirical analysis of the Twin Prime Index Conjecture, improving on the original version of this script. This conjecture reframes the search for twin primes into a search for a specific set of integers, k, which generate twin prime pairs of the form (6k-1, 6k+1). The script automates the discovery of these indices and, more importantly, implements a sophisticated statistical method to test the validity of the conjecture against established number theory.

To achieve this, the script incorporates a linearization and regression technique based on the prior manual, AI-guided empirical analysis. While a simple calculation of the density of indices provides a baseline, it is insufficient for a robust conclusion due to statistical noise and slow convergence. The linearization method transforms the raw density data into a linear model, allowing for a more accurate and stable extrapolation of the asymptotic behavior of the indices. This statistical forecast provides a much stronger piece of evidence than a simple point-in-time measurement.

Furthermore, the script’s methodology is designed to test a specific prediction derived from the First Hardy-Littlewood Conjecture. The standard conjecture predicts the density of twin primes p, p+2. By adapting this for primes of the form 6k±1, the theoretical density constant scales from 2*C₂ to 12*C₂, where C₂ is the twin prime constant (≈0.66). The script’s final extrapolated value is therefore a direct empirical test of this scaled constant, K ≈ 7.92, providing a quantitative measure of the Twin Prime Index Conjecture’s accuracy.

2. Background and Methodology

2.1. The Twin Prime Index Conjecture

The script is based on the theory that an integer k is the index of a twin prime pair (6k-1, 6k+1) if and only if it cannot be generated by the Diophantine formula k = |6xy + x + y| for any non-zero integers x and y.

The set of all “creatable” integers is defined as K_composite = {|6xy + x + y| : x,y ∈ ℤ \ {0}}. The set of twin prime indices is therefore the complement of this set in the positive integers, ℤ⁺ \ K_composite.

2.2. Analytical Approach

The script performs two primary functions:

1. Sieve for K_composite: It programmatically generates values belonging to K_composite up to a specified upper bound K_LIMIT and identifies the integers that were not generated.
2. Density Analysis: It calculates the density of the uncreatable integers and uses a linearization technique based on the Hardy-Littlewood conjecture to extrapolate an empirical value for the constant K ≈ 12 * C₂ ≈ 7.92.

3. Prerequisites

• Python 3.6+
• NumPy
• scikit-learn

4. Installation

The required libraries can be installed using pip:

pip install numpy scikit-learn

5. Script Architecture and Components

The script consists of four primary functions and a main execution block.

• find_uncreatable_numbers(max_k: int)
  • Purpose: Identifies all integers from 1 to max_k that do not belong to the set K_composite.
  • Methodology: A brute-force sieve that iterates through a bounded range of non-zero integer pairs (x, y). It calculates k = abs(6*x*y + x + y) for each pair and adds the result to a set of generated values. The function returns the sorted list of integers that are not present in this set.
• analyze_uncreatable_density(max_k: int)
  • Purpose: Processes the list of uncreatable numbers to generate density data.
  • Methodology: Takes the list of uncreatable numbers [n_1, n_2, n_3, …] and creates a corresponding list of density fractions [“1/n_1”, “2/n_2”, “3/n_3”, …], where the numerator c is the cumulative count of the index n.
• perform_linearization_analysis(density_annotations: List[str])
  • Purpose: Calculates an extrapolated value for the constant K using linear regression.
  • Methodology: This function tests the theoretical relationship Z(n) ≈ K + D/ln(6n), where Z(n) = (c/n)*(ln(6n))². By plotting Y = 1/ln(6n) vs. X = Z(n), the relationship becomes linear: Y ≈ (1/D) * (X – K). The script fits a standard linear model Y = mX + b to the full set of generated data points (excluding a few initial noisy points). The x-intercept of this line (-b/m) provides a robust, extrapolated estimate for the constant K.
• Main Execution Block (if __name__ == “__main__”:)
  • Purpose: Orchestrates the analysis and presents the final results.
  • Workflow:
    1. Sets the global K_LIMIT.
    2. Executes the sieve and density analysis.
    3. Prints a summary of the found data.
    4. Calls perform_linearization_analysis to compute the primary result.
    5. Presents both a simple point-estimate (calculated from the last data point) and the more robust extrapolated value for comparison.

6. Interpretation of Results and Observed Behavior

The script produces two key metrics for the Hardy-Littlewood constant K. Their behavior and reliability evolve as the analysis limit (K_LIMIT) is increased.

1. Point-Estimate Constant: This value is calculated using only the final uncreatable number n and its count c. It is a direct measurement of the test statistic Z(n) = (c/n)*(log n)² at the maximum limit of the analysis. While informative, it is highly susceptible to local fluctuations in data density.
2. Extrapolated Constant (from Linearization): This is the primary result of the analysis. By fitting a linear model to the trend of the entire dataset, this method mitigates the effect of local noise and provides a statistical forecast of the true asymptotic value of K.

Observed Behavior Across Different K_LIMITs

Analysis performed at increasing orders of magnitude (100, 1,000, 10,000, etc.) reveals a clear and consistent pattern of convergence for the extrapolated constant:

• At K_LIMIT = 100 and 1,000: The dataset is too small and dominated by statistical noise and small-number irregularities. The linearization analysis is unreliable, producing extrapolated values (-9.27 and 2.84, respectively) that are not meaningful.
• At K_LIMIT = 10,000: The method begins to produce a sensible result. The extrapolated K (~7.20) is a significant improvement and demonstrates that the underlying trend is starting to emerge from the noise.
• At K_LIMIT = 100,000 and 1,000,000: The analysis reaches a high degree of accuracy. The extrapolated values (~7.48 and ~7.82) are substantially closer to the theoretical constant (~7.92) than their corresponding point-estimates. The run at 1,000,000 was observed to produce a result with approximately 98.7% accuracy.
• At K_LIMIT > 1,000,000: It has been observed that the accuracy of the extrapolation can show minor oscillations. For example, the results for 10,000,000 and 100,000,000 (~7.57) were slightly less accurate than the 1,000,000 run. This is a known phenomenon in empirical number theory, likely caused by large-scale variations in the density of primes (akin to Chebyshev’s bias) that can temporarily influence the local trend of the regression line.

Conclusion:
The extrapolated constant is the more robust and predictive of the two metrics. While its accuracy can fluctuate slightly at very large scales, it consistently provides a better approximation of the theoretical limit than the point-estimate. The data strongly supports the conclusion that the linearization method is effective and that the Twin Prime Index Conjecture aligns with the predictions of the Hardy-Littlewood conjecture.

7. Known Limitations and Future Optimizations

The script is a functional tool for empirical analysis but has known limitations that provide avenues for future work.

• Performance: The core find_uncreatable_numbers function is a brute-force search. Its runtime scales significantly as K_LIMIT increases, making analysis beyond ~10^8 impractical on standard hardware.
• Potential Optimizations:
  1. Parallel Processing: The search loop is highly parallelizable. Using Python’s multiprocessing module to distribute the search space for x across multiple CPU cores would provide a near-linear performance improvement.
  2. Memory Optimization: For extremely large K_LIMITs, the generated_k_values set can become memory-intensive. This can be mitigated by using a NumPy boolean array as a bitmask, which has a much smaller memory footprint.
  3. Algorithmic Sieve: A more advanced implementation could replace the brute-force search with a direct mathematical test based on the algebraic consequences of the generating formula (i.e., 6k+1 = (6x+1)(6y+1) and 6k-1 = -(6x+1)(6y+1)). This would provide a more efficient method for identifying members of K_composite while preserving the theoretical framework of the conjecture.
  4. Visualization Tools: Adding visualizations could enhance the understanding of the asymptotic behavior of this Diophantine complement of the composites. (Example artifact I created with Claude AI : Twin Prime Density Analyzer.)

Density Analyzer Script 2.0:

"""
Density Sequence Analyzer for Twin Prime Indices

This script investigates the distribution of twin primes by analyzing the Diophantine
equation |6xy + x + y| = k. Based on the conjecture that a number 'k' is the index
of a twin prime pair (6k-1, 6k+1) if and only if it CANNOT be generated by the
expression, this code performs the following analysis:

1.  **Finds "Uncreatable" Numbers**: It identifies all integers up to a specified
    limit (K_LIMIT) that are not solutions to the equation for any non-zero
    integers x and y. These are the twin prime indices.

2.  **Calculates the Largest Twin Prime Pair**: Using the largest uncreatable
    index 'k' found, it calculates the corresponding twin prime pair using the
    formula (6k - 1, 6k + 1).

3.  **Computes the Empirical Hardy-Littlewood Constant**: It calculates the density
    of the uncreatable numbers and uses this to compute an empirical value for a
    constant related to the Hardy-Littlewood twin prime conjecture. This script
    tests the prediction that the value (c/n) * (log n)² should converge to
    12 * C₂, where C₂ is the twin prime constant.

4.  **Performs Linearization Analysis**: To get a more robust estimate of the
    constant, it linearizes the data by plotting 1/log(6n) vs. Z(n) where
    Z(n) = (c/n)*(log(6n))**2. The x-intercept of the resulting linear
    regression provides a data-driven, extrapolated estimate for the constant.
"""

from typing import List, Dict, Set, Union
import numpy as np
from sklearn.linear_model import LinearRegression

def find_uncreatable_numbers(max_k: int) -> List[int]:
    """
    Finds all integers up to a limit that cannot be expressed by |6xy + x + y|.
    """
    if not isinstance(max_k, int) or max_k < 1:
        raise ValueError("max_k must be an integer greater than 0.")

    generated_k_values: Set[int] = set()
    x_limit = (max_k // 5) + 2

    for x in range(-x_limit, x_limit):
        if x == 0: continue
        den = 6 * x + 1
        if den > 0:
            y_lower, y_upper = (-max_k - x) / den, (max_k - x) / den
        else:
            y_lower, y_upper = (max_k - x) / den, (-max_k - x) / den

        for y in range(int(y_lower), int(y_upper) + 1):
            if y == 0: continue
            k_val = abs(6 * x * y + x + y)
            if 0 < k_val <= max_k:
                generated_k_values.add(k_val)

    uncreatable_set = set(range(1, max_k + 1)) - generated_k_values
    return sorted(list(uncreatable_set))

def analyze_uncreatable_density(max_k: int) -> Dict[str, Union[List[int], List[str]]]:
    """
    Finds uncreatable numbers and annotates them with their density fractions.
    """
    uncreatable_list = find_uncreatable_numbers(max_k)
    annotated_list = [f"{i+1}/{n}" for i, n in enumerate(uncreatable_list)]
    return {"complement_set": uncreatable_list, "annotated_density": annotated_list}

def get_largest_twin_prime(last_uncreatable_k: int) -> str:
    """
    Calculates the twin prime pair for a given index 'k'.
    """
    prime1 = 6 * last_uncreatable_k - 1
    prime2 = 6 * last_uncreatable_k + 1
    return f"{prime1},{prime2}"

def perform_linearization_analysis(density_annotations: List[str]) -> Dict[str, float]:
    """
    Performs a linear regression on transformed data to extrapolate the constant K.

    This function calculates Z(n) = (c/n)*(log(6n))**2 and Y_plot = 1/log(6n).
    It then fits a linear model: Y_plot = m*Z(n) + b. The x-intercept of this
    line (-b/m) provides a robust estimate of the asymptotic constant K.

    Args:
        density_annotations: A list of 'c/n' strings.

    Returns:
        A dictionary with the extrapolated constant, slope, and intercept.
    """
    z_values = []
    y_plot_values = []

    # Start from the 10th data point to avoid noise from very small n
    for item in density_annotations[10:]:
        c_str, n_str = item.split('/')
        c, n = int(c_str), int(n_str)
        
        # As per the theory, use log(6n) for the Z-value
        z_val = (c / n) * (np.log(6 * n) ** 2)
        y_plot_val = 1 / np.log(6 * n)
        
        z_values.append(z_val)
        y_plot_values.append(y_plot_val)

    if len(z_values) < 2:
        return {}

    # Reshape X for scikit-learn
    X = np.array(z_values).reshape(-1, 1)
    y = np.array(y_plot_values)

    model = LinearRegression()
    model.fit(X, y)

    m = model.coef_[0]
    b = model.intercept_

    if m == 0: # Avoid division by zero
        return {}
        
    extrapolated_K = -b / m

    return {"extrapolated_K": extrapolated_K, "slope": m, "intercept": b}


# --- Main Execution Block ---
if __name__ == "__main__":
    K_LIMIT = 1000000

    print(f"--- Diophantine Analysis for |6xy + x + y| up to k = {K_LIMIT} ---")

    try:
        analysis_results = analyze_uncreatable_density(K_LIMIT)
        uncreatable_numbers = analysis_results["complement_set"]
        density_annotations = analysis_results["annotated_density"]
        
        num_found = len(uncreatable_numbers)
        print(f"\nFound {num_found} uncreatable integers up to {K_LIMIT}.")
        
        if num_found > 40:
            print("\n--- Uncreatable Integers (First 20 and Last 20) ---")
            print(f"First 20: {uncreatable_numbers[:20]}")
            print("...")
            print(f"Last 20:  {uncreatable_numbers[-20:]}")

            print("\n--- Annotated Density Fractions (First 20 and Last 20) ---")
            print(f"First 20: {density_annotations[:20]}")
            print("...")
            print(f"Last 20:  {density_annotations[-20:]}")

        # --- Final Analysis Summary ---
        if density_annotations:
            # --- Point-Estimate Calculation (based on last value) ---
            print("\n--- Point-Estimate Constant (based on last value in sequence) ---")
            last_item = density_annotations[-1]
            c_str, n_str = last_item.split('/')
            c, n = int(c_str), int(n_str)
            
            if n > 1:
                largest_twin_prime_pair = get_largest_twin_prime(n)
                print(f"Last uncreatable number found (n): {n}")
                print(f"Largest Twin Prime Pair Found: {largest_twin_prime_pair}")
                
                # Note: Original script used (log n)**2, refined theory uses (log(6n))**2
                empirical_K_point_estimate = (c / n) * (np.log(n) ** 2)
                print(f"Point-Estimate K = (c/n) * (log n)²: {empirical_K_point_estimate:.6f}")
            
            # --- Linearization Analysis ---
            print("\n--- Linearization Analysis (extrapolated from all data) ---")
            linearization_results = perform_linearization_analysis(density_annotations)
            
            if linearization_results:
                extrapolated_K = linearization_results["extrapolated_K"]
                m = linearization_results["slope"]
                b = linearization_results["intercept"]
                C2 = 0.6601618158468696
                theoretical_K = 12 * C2
                
                print(f"Regression model: 1/ln(6n) = {m:.4f} * Z(n) + {b:.4f}")
                print(f"Extrapolated Constant K (from x-intercept): {extrapolated_K:.6f}")
                print(f"Theoretical Constant K = 12 * C₂:           {theoretical_K:.6f}")
                print(f"Difference (Theoretical - Extrapolated):    {theoretical_K - extrapolated_K:.6f}")
            else:
                print("Not enough data to perform linearization analysis.")
        else:
            print("No uncreatable numbers found.")

    except ValueError as e:
        print(f"Error: {e}")

Example Outputs:

— Diophantine Analysis for |6xy + x + y| up to k = 1 —

Found 1 uncreatable integers up to 1.

— Point-Estimate Constant (based on last value in sequence) —

— Linearization Analysis (extrapolated from all data) —
Not enough data to perform linearization analysis.
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 10 —

Found 6 uncreatable integers up to 10.

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 10
Largest Twin Prime Pair Found: 59,61
Point-Estimate K = (c/n) * (log n)²: 3.181139

— Linearization Analysis (extrapolated from all data) —
Not enough data to perform linearization analysis.
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 100 —

Found 26 uncreatable integers up to 100.

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 100
Largest Twin Prime Pair Found: 599,601
Point-Estimate K = (c/n) * (log n)²: 5.513974

— Linearization Analysis (extrapolated from all data) —
Regression model: 1/ln(6n) = 0.0085 * Z(n) + 0.0788
Extrapolated Constant K (from x-intercept): -9.274286
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical – Extrapolated): 17.196227
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 1000 —

Found 142 uncreatable integers up to 1000.

— Uncreatable Integers (First 20 and Last 20) —
First 20: [1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58]
…
Last 20: [800, 822, 828, 835, 837, 850, 872, 880, 903, 907, 913, 917, 920, 940, 942, 943, 957, 975, 978, 980]

— Annotated Density Fractions (First 20 and Last 20) —
First 20: [‘1/1’, ‘2/2’, ‘3/3’, ‘4/5’, ‘5/7’, ‘6/10’, ‘7/12’, ‘8/17’, ‘9/18′, ’10/23′, ’11/25′, ’12/30′, ’13/32′, ’14/33′, ’15/38′, ’16/40′, ’17/45′, ’18/47′, ’19/52′, ’20/58’]
…
Last 20: [‘123/800’, ‘124/822’, ‘125/828’, ‘126/835’, ‘127/837’, ‘128/850’, ‘129/872’, ‘130/880’, ‘131/903’, ‘132/907’, ‘133/913’, ‘134/917’, ‘135/920’, ‘136/940’, ‘137/942’, ‘138/943’, ‘139/957’, ‘140/975’, ‘141/978’, ‘142/980’]

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 980
Largest Twin Prime Pair Found: 5879,5881
Point-Estimate K = (c/n) * (log n)²: 6.873725

— Linearization Analysis (extrapolated from all data) —
Regression model: 1/ln(6n) = 0.0169 * Z(n) + -0.0481
Extrapolated Constant K (from x-intercept): 2.839589
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical – Extrapolated): 5.082352
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 10000 —

Found 810 uncreatable integers up to 10000.

— Uncreatable Integers (First 20 and Last 20) —
First 20: [1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58]
…
Last 20: [9695, 9705, 9728, 9732, 9740, 9742, 9767, 9798, 9818, 9835, 9837, 9842, 9868, 9870, 9893, 9903, 9907, 9912, 9938, 9945]

— Annotated Density Fractions (First 20 and Last 20) —
First 20: [‘1/1’, ‘2/2’, ‘3/3’, ‘4/5’, ‘5/7’, ‘6/10’, ‘7/12’, ‘8/17’, ‘9/18′, ’10/23′, ’11/25′, ’12/30′, ’13/32′, ’14/33′, ’15/38′, ’16/40′, ’17/45′, ’18/47′, ’19/52′, ’20/58’]
…
Last 20: [‘791/9695’, ‘792/9705’, ‘793/9728’, ‘794/9732’, ‘795/9740’, ‘796/9742’, ‘797/9767’, ‘798/9798’, ‘799/9818’, ‘800/9835’, ‘801/9837’, ‘802/9842’, ‘803/9868’, ‘804/9870’, ‘805/9893’, ‘806/9903’, ‘807/9907’, ‘808/9912’, ‘809/9938’, ‘810/9945’]

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 9945
Largest Twin Prime Pair Found: 59669,59671
Point-Estimate K = (c/n) * (log n)²: 6.900989

— Linearization Analysis (extrapolated from all data) —
Regression model: 1/ln(6n) = 0.0353 * Z(n) + -0.2539
Extrapolated Constant K (from x-intercept): 7.200925
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical – Extrapolated): 0.721017
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 100000 —

Found 5330 uncreatable integers up to 100000.

— Uncreatable Integers (First 20 and Last 20) —
First 20: [1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58]
…
Last 20: [99488, 99522, 99545, 99568, 99587, 99612, 99613, 99628, 99650, 99675, 99698, 99748, 99775, 99788, 99822, 99837, 99858, 99913, 99950, 99990]

— Annotated Density Fractions (First 20 and Last 20) —
First 20: [‘1/1’, ‘2/2’, ‘3/3’, ‘4/5’, ‘5/7’, ‘6/10’, ‘7/12’, ‘8/17’, ‘9/18′, ’10/23′, ’11/25′, ’12/30′, ’13/32′, ’14/33′, ’15/38′, ’16/40′, ’17/45′, ’18/47′, ’19/52′, ’20/58’]
…
Last 20: [‘5311/99488’, ‘5312/99522’, ‘5313/99545’, ‘5314/99568’, ‘5315/99587’, ‘5316/99612’, ‘5317/99613’, ‘5318/99628’, ‘5319/99650’, ‘5320/99675’, ‘5321/99698’, ‘5322/99748’, ‘5323/99775’, ‘5324/99788’, ‘5325/99822’, ‘5326/99837’, ‘5327/99858’, ‘5328/99913’, ‘5329/99950’, ‘5330/99990’]

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 99990
Largest Twin Prime Pair Found: 599939,599941
Point-Estimate K = (c/n) * (log n)²: 7.065363

— Linearization Analysis (extrapolated from all data) —
Regression model: 1/ln(6n) = 0.0386 * Z(n) + -0.2890
Extrapolated Constant K (from x-intercept): 7.481583
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical – Extrapolated): 0.440358
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 1000000 —

Found 37915 uncreatable integers up to 1000000.

— Uncreatable Integers (First 20 and Last 20) —
First 20: [1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58]
…
Last 20: [999527, 999537, 999558, 999560, 999570, 999602, 999640, 999673, 999680, 999787, 999812, 999862, 999868, 999877, 999885, 999927, 999938, 999955, 999985, 999987]

— Annotated Density Fractions (First 20 and Last 20) —
First 20: [‘1/1’, ‘2/2’, ‘3/3’, ‘4/5’, ‘5/7’, ‘6/10’, ‘7/12’, ‘8/17’, ‘9/18′, ’10/23′, ’11/25′, ’12/30′, ’13/32′, ’14/33′, ’15/38′, ’16/40′, ’17/45′, ’18/47′, ’19/52′, ’20/58’]
…
Last 20: [‘37896/999527’, ‘37897/999537’, ‘37898/999558’, ‘37899/999560’, ‘37900/999570’, ‘37901/999602’, ‘37902/999640’, ‘37903/999673’, ‘37904/999680’, ‘37905/999787’, ‘37906/999812’, ‘37907/999862’, ‘37908/999868’, ‘37909/999877’, ‘37910/999885’, ‘37911/999927’, ‘37912/999938’, ‘37913/999955’, ‘37914/999985’, ‘37915/999987’]

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 999987
Largest Twin Prime Pair Found: 5999921,5999923
Point-Estimate K = (c/n) * (log n)²: 7.236853

— Linearization Analysis (extrapolated from all data) —
Regression model: 1/ln(6n) = 0.0451 * Z(n) + -0.3525
Extrapolated Constant K (from x-intercept): 7.821685
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical – Extrapolated): 0.100257
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 10000000 —

Found 280557 uncreatable integers up to 10000000.

— Uncreatable Integers (First 20 and Last 20) —
First 20: [1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58]
…
Last 20: [9999327, 9999353, 9999370, 9999462, 9999523, 9999525, 9999542, 9999575, 9999593, 9999638, 9999682, 9999685, 9999755, 9999808, 9999880, 9999883, 9999938, 9999973, 9999980, 9999997]

— Annotated Density Fractions (First 20 and Last 20) —
First 20: [‘1/1’, ‘2/2’, ‘3/3’, ‘4/5’, ‘5/7’, ‘6/10’, ‘7/12’, ‘8/17’, ‘9/18′, ’10/23′, ’11/25′, ’12/30′, ’13/32′, ’14/33′, ’15/38′, ’16/40′, ’17/45′, ’18/47′, ’19/52′, ’20/58’]
…
Last 20: [‘280538/9999327’, ‘280539/9999353’, ‘280540/9999370’, ‘280541/9999462’, ‘280542/9999523’, ‘280543/9999525’, ‘280544/9999542’, ‘280545/9999575’, ‘280546/9999593’, ‘280547/9999638’, ‘280548/9999682’, ‘280549/9999685’, ‘280550/9999755’, ‘280551/9999808’, ‘280552/9999880’, ‘280553/9999883’, ‘280554/9999938’, ‘280555/9999973’, ‘280556/9999980’, ‘280557/9999997’]

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 9999997
Largest Twin Prime Pair Found: 59999981,59999983
Point-Estimate K = (c/n) * (log n)²: 7.288677

— Linearization Analysis (extrapolated from all data) —
Regression model: 1/ln(6n) = 0.0389 * Z(n) + -0.2943
Extrapolated Constant K (from x-intercept): 7.573665
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical – Extrapolated): 0.348277
Press any key to continue . . .

— Diophantine Analysis for |6xy + x + y| up to k = 100000000 —

Found 2166300 uncreatable integers up to 100000000.

— Uncreatable Integers (First 20 and Last 20) —
First 20: [1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58]
…
Last 20: [99998997, 99999032, 99999065, 99999067, 99999085, 99999147, 99999177, 99999230, 99999310, 99999368, 99999415, 99999478, 99999533, 99999585, 99999620, 99999653, 99999662, 99999823, 99999842, 99999905]

— Annotated Density Fractions (First 20 and Last 20) —
First 20: [‘1/1’, ‘2/2’, ‘3/3’, ‘4/5’, ‘5/7’, ‘6/10’, ‘7/12’, ‘8/17’, ‘9/18′, ’10/23′, ’11/25′, ’12/30′, ’13/32′, ’14/33′, ’15/38′, ’16/40′, ’17/45′, ’18/47′, ’19/52′, ’20/58’]
…
Last 20: [‘2166281/99998997’, ‘2166282/99999032’, ‘2166283/99999065’, ‘2166284/99999067’, ‘2166285/99999085’, ‘2166286/99999147’, ‘2166287/99999177’, ‘2166288/99999230’, ‘2166289/99999310’, ‘2166290/99999368’, ‘2166291/99999415’, ‘2166292/99999478’, ‘2166293/99999533’, ‘2166294/99999585’, ‘2166295/99999620’, ‘2166296/99999653’, ‘2166297/99999662’, ‘2166298/99999823’, ‘2166299/99999842’, ‘2166300/99999905’]

— Point-Estimate Constant (based on last value in sequence) —
Last uncreatable number found (n): 99999905
Largest Twin Prime Pair Found: 599999429,599999431
Point-Estimate K = (c/n) * (log n)²: 7.350727

— Linearization Analysis (extrapolated from all data) —
Regression model: 1/ln(6n) = 0.0391 * Z(n) + -0.2961
Extrapolated Constant K (from x-intercept): 7.578705
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical – Extrapolated): 0.343237
Press any key to continue . . .

Disclosure: This blog was produced with assistance aistudio.google.com