The following Python code is intended to calculate the density of each twin prime pair as a relative value in the sequence of all possible twin prime pairs 6k-1,6k+1. (An extension of the empirical investigation here which corroborated the Hardy Littlewood Conjecture 1 by finding an empirical value of 7.53 compared to the conjectured 12*0.66=7.92 using a limited data set of k indices.)
Note: If you want to include 3,5 for a complete count, you can just add 1/1 to the value you consider in the c/n fraction list, so the first value in the below would be 2/2 (as opposed to 1/1).
Recall our reformulation of the Twin Prime Conjecture: Twin Prime Index Conjecture
Let:f(x,y)=∣6xy+x+y∣
where 𝑥,𝑦 ∈ 𝑍∖{0} (i.e., both are non-zero integers, so may be positive or negative).
Define the set:
𝐾composite = {𝑓(𝑥,𝑦): 𝑥≠0, 𝑦≠0}
Then: A positive integer 𝑘 is the index of a twin prime pair (6𝑘−1,6𝑘+1) if and only if:
𝑘∉𝐾composite
Therefore, the Twin Prime Conjecture is true if and only if:
𝑍+∖𝐾composite is infinite
In plain language: There are infinitely many twin primes if and only if there are infinitely many positive integers 𝑘 that cannot be written in the form ∣6𝑥𝑦+𝑥+𝑦∣ for any non-zero integers 𝑥,𝑦.
Decoding the c/n Fraction: A Measure of Twin Prime Density
Our Python script generates a list of fractions in the format c/n. While simple in appearance, each fraction is a data point in the study of the Twin Prime Conjecture. Let’s break down what c and n represent.
What are n and c?
First, n is one of the “uncreatable” integers our script finds. These are special numbers because they are the keys to generating twin primes of the form (6k-1, 6k+1).
- n is the Twin Prime Index: When our code produces the fraction 5/7, the denominator n = 7 is an integer that cannot be created by the formula |6xy + x + y|. This means k=7 generates a twin prime pair:
- 6n – 1 = 6(7) – 1 = 41 (a prime)
- 6n + 1 = 6(7) + 1 = 43 (a prime)
Next, c is the rank of that index in the sequence of all such indices found so far.
- c is the Cumulative Count: For the fraction 5/7, the numerator c = 5 tells us that 7 is the 5th such special number we have discovered. The sequence of these indices begins: 1, 2, 3, 5, 7, 10, …
Introducing a Counting Function: π_twin_k(N)
To analyze density formally, we define a counting function for our sequence:
Let π_twin_k(N) be the function that counts the number of “uncreatable” integers k that are less than or equal to N.
In our c/n notation, the relationship is simple: c = π_twin_k(n). Therefore, the fraction our code calculates is a precise measure of density:
Density at n = c / n = π_twin_k(n) / n
What This Density Reveals: The Hardy-Littlewood Conjecture
The real power of this analysis comes from comparing our results to the famous First Hardy-Littlewood Conjecture. This conjecture doesn’t just say there are infinite twin primes; it predicts their exact density.
1. The Standard Prediction
The “textbook” version of the conjecture is for the number of primes p (where p ≤ X) such that p+2 is also prime. Let’s call this count π₂(X). The formula is:
π₂(X) ≈ 2 * C₂ * (X / (log X)²)
Here, C₂ is the twin prime constant, approximately 0.66016.
2. Adapting the Formula for Our Sequence
Our script doesn’t count primes p, it counts indices k up to a limit n. To use the formula, we must connect our (k, n) world to the standard (p, X) world.
The crucial link is that our primes are in two arithmetic progressions of the form p = |6k + 1|. This means the primes we are finding extend up to a value of approximately 6n. Therefore, we should substitute X = 6n into the standard formula.
c = π_twin_k(n) ≈ π₂(6n)
c ≈ 2 * C₂ * ( 6n / (log(6n))² )
Now we have a prediction for our count, c.
3. Deriving Our Theoretical Constant K
Our analysis script checked the behavior of the product (c/n) * (log n)². Let’s see what the adapted formula predicts for this value:
First, let’s calculate the predicted density c/n:
c/n ≈ [ 2 * C₂ * (6n / (log(6n))²) ] / n
c/n ≈ 12 * C₂ / (log(6n))²
Now, multiply by (log n)² just as the script did:
K ≈ [ 12 * C₂ / (log(6n))² ] * (log n)²
Using the logarithm property log(6n) = log(6) + log(n), we get:
K ≈ 12 * C₂ * [ (log n)² / (log(6) + log(n))² ]
As n gets very large, the log(n) term dominates the constant log(6), and the fraction (log n)² / (log(6) + log(n))² gets closer and closer to 1.
This leaves us with a stunningly clear prediction for our constant K:
K ≈ 12 * C₂
4. The Final Result
Plugging in the value for the twin prime constant gives us our answer:
K ≈ 12 * 0.6601618… ≈ 7.922
This explains why the script’s calculation was converging to a value near 7, not the 2 * C₂ ≈ 1.32 that a naive application of the formula would suggest. The factor of 6 in our prime-generating formula (6k±1) scales the final constant by 6, turning 2C₂ into 12C₂.
The script’s empirical result (K ≈ 7.360819 at n=150,000,000) acts as a powerful verification of this mathematical reasoning.
Python Code:
"""
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.
The script outputs truncated lists of the found numbers and their density fractions,
followed by a final summary of the analysis.
"""
from typing import List, Dict, Set, Union
import numpy as np
def find_uncreatable_numbers(max_k: int) -> List[int]:
"""
Finds all integers up to a limit that cannot be expressed by |6xy + x + y|.
This function iterates through a bounded range of non-zero integer pairs (x, y)
and calculates k = |6xy + x + y|. It stores these "creatable" values in a set.
Finally, it returns the sorted list of numbers from 1 to max_k that were
never created.
Args:
max_k: The upper integer limit (inclusive) for the analysis.
Returns:
A sorted list of "uncreatable" integers, which are the indices of
twin prime pairs of the form (6k-1, 6k+1).
"""
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()
# The search range for x and y is bounded. For |6xy+x+y| <= max_k, the
# magnitude of x and y are inversely related. We can establish a conservative
# limit for the search by solving for y.
x_limit = (max_k // 5) + 2
for x in range(-x_limit, x_limit):
if x == 0:
continue # x and y must be non-zero.
den = 6 * x + 1
# Derive bounds for y from the inequality: -max_k <= y(6x+1)+x <= max_k
if den > 0:
y_lower = (-max_k - x) / den
y_upper = (max_k - x) / den
else: # den < 0
y_lower = (max_k - x) / den
y_upper = (-max_k - x) / den
# Iterate through all valid integer values for y
for y in range(int(y_lower), int(y_upper) + 1):
if y == 0:
continue # x and y must be non-zero.
k_val = abs(6 * x * y + x + y)
if 0 < k_val <= max_k:
generated_k_values.add(k_val)
# The full set of integers we are checking against.
all_integers = set(range(1, max_k + 1))
# The complement is the set of integers that were never generated.
uncreatable_set = all_integers - 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.
This function calls `find_uncreatable_numbers` and then processes the
resulting list. Each uncreatable number 'n' is paired with its rank 'c'
in the sequence, producing a list of fractions 'c/n'.
Args:
max_k: The upper integer limit for the analysis.
Returns:
A dictionary containing the list of uncreatable integers and the list
of their corresponding density fractions.
"""
uncreatable_list = find_uncreatable_numbers(max_k)
# Create the 'c/n' annotations. 'c' is the cumulative count (1-based index).
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'.
The twin primes are generated using the formula (6k - 1, 6k + 1).
Args:
last_uncreatable_k: The twin prime index 'k' from the analysis.
Returns:
A string formatted as "prime1,prime2".
"""
prime1 = 6 * last_uncreatable_k - 1
prime2 = 6 * last_uncreatable_k + 1
return f"{prime1},{prime2}"
# --- Main Execution Block ---
if __name__ == "__main__":
# Set the upper limit for 'k' for the analysis.
K_LIMIT = 1500000
print(f"--- Diophantine Analysis for |6xy + x + y| up to k = {K_LIMIT} ---")
try:
# Run the core analysis to find uncreatable numbers and their densities.
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}.")
# To keep the output readable, display only the first and last 20 results.
if num_found > 0:
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 ---
print("\n--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---")
if density_annotations:
# Get the last item from the list, e.g., '53866/1499983'
last_item = density_annotations[-1]
c_str, n_str = last_item.split('/')
c = int(c_str) # Cumulative count of uncreatable numbers
n = int(n_str) # The last (and largest) uncreatable number found
if n > 1:
# Calculate the largest twin prime pair from the largest index 'n'.
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}")
print(f"Cumulative count of uncreatable numbers (c): {c}")
print(f"Final Density (c/n): {c/n:.6f}")
# This is the core calculation for the empirical constant.
# It evaluates K = (c/n) * (log n)², which should converge to 12*C₂.
empirical_K = (c / n) * (np.log(n) ** 2)
# The theoretical constant for comparison.
C2 = 0.6601618158468696 # Twin prime constant
theoretical_K = 12 * C2
print(f"\nEmpirical Constant K = (c/n) * (log n)²: {empirical_K:.6f}")
print(f"Theoretical Constant K = 12 * C₂: {theoretical_K:.6f}")
print(f"Difference (Theoretical - Empirical): {theoretical_K - empirical_K:.6f}")
else:
print("Cannot calculate constant for n <= 1.")
else:
print("No uncreatable numbers found.")
except ValueError as e:
print(f"Error: {e}")Example Outputs :
10:
--- Diophantine Analysis for |6xy + x + y| up to k = 10 ---
Found 6 uncreatable integers up to 10.
--- Uncreatable Integers (First 20 and Last 20) ---
First 20: [1, 2, 3, 5, 7, 10]
...
Last 20: [1, 2, 3, 5, 7, 10]
--- Annotated Density Fractions (First 20 and Last 20) ---
First 20: ['1/1', '2/2', '3/3', '4/5', '5/7', '6/10']
...
Last 20: ['1/1', '2/2', '3/3', '4/5', '5/7', '6/10']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 10
Largest Twin Prime Pair Found: 59,61
Cumulative count of uncreatable numbers (c): 6
Final Density (c/n): 0.600000
Empirical Constant K = (c/n) * (log n)²: 3.181139
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 4.740803
Press any key to continue . . .
100:
--- Diophantine Analysis for |6xy + x + y| up to k = 100 ---
Found 26 uncreatable integers up to 100.
--- 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: [12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58, 70, 72, 77, 87, 95, 100]
--- 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: ['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', '21/70', '22/72', '23/77', '24/87', '25/95', '26/100']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 100
Largest Twin Prime Pair Found: 599,601
Cumulative count of uncreatable numbers (c): 26
Final Density (c/n): 0.260000
Empirical Constant K = (c/n) * (log n)²: 5.513974
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 2.407968
Press any key to continue . . .
1,000:
--- 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']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 980
Largest Twin Prime Pair Found: 5879,5881
Cumulative count of uncreatable numbers (c): 142
Final Density (c/n): 0.144898
Empirical Constant K = (c/n) * (log n)²: 6.873725
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 1.048217
Press any key to continue . . .
10,000:
--- 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']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 9945
Largest Twin Prime Pair Found: 59669,59671
Cumulative count of uncreatable numbers (c): 810
Final Density (c/n): 0.081448
Empirical Constant K = (c/n) * (log n)²: 6.900989
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 1.020953
Press any key to continue . . .
100,000:
--- 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']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 99990
Largest Twin Prime Pair Found: 599939,599941
Cumulative count of uncreatable numbers (c): 5330
Final Density (c/n): 0.053305
Empirical Constant K = (c/n) * (log n)²: 7.065363
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 0.856579
Press any key to continue . . .
1,000,000:
--- 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']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 999987
Largest Twin Prime Pair Found: 5999921,5999923
Cumulative count of uncreatable numbers (c): 37915
Final Density (c/n): 0.037915
Empirical Constant K = (c/n) * (log n)²: 7.236853
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 0.685089
Press any key to continue . . .
10,000,000:
--- 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']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 9999997
Largest Twin Prime Pair Found: 59999981,59999983
Cumulative count of uncreatable numbers (c): 280557
Final Density (c/n): 0.028056
Empirical Constant K = (c/n) * (log n)²: 7.288677
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 0.633265
Press any key to continue . . .
100,000,000:
--- 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']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 99999905
Largest Twin Prime Pair Found: 599999429,599999431
Cumulative count of uncreatable numbers (c): 2166300
Final Density (c/n): 0.021663
Empirical Constant K = (c/n) * (log n)²: 7.350727
Theoretical Constant K = 12 * C₂: 7.921942
Difference (Theoretical - Empirical): 0.571214
Press any key to continue . . .
150,000,000:
--- Diophantine Analysis for |6xy + x + y| up to k = 150000000 ---
Found 3115261 uncreatable integers up to 150000000.
--- 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: [149999097, 149999118, 149999138, 149999152, 149999253, 149999362, 149999365, 149999377, 149999413, 149999453, 149999570, 149999598, 149999623, 149999630, 149999675, 149999712, 149999777, 149999845, 149999923, 149999957]
--- 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: ['3115242/149999097', '3115243/149999118', '3115244/149999138', '3115245/149999152', '3115246/149999253', '3115247/149999362', '3115248/149999365', '3115249/149999377', '3115250/149999413', '3115251/149999453', '3115252/149999570', '3115253/149999598', '3115254/149999623', '3115255/149999630', '3115256/149999675', '3115257/149999712', '3115258/149999777', '3115259/149999845', '3115260/149999923', '3115261/149999957']
--- Hardy-Littlewood Empirical Constant (for last value in sequence) ---
Last uncreatable number found (n): 149999957
Largest Twin Prime Pair Found: 899999741,899999743
Cumulative count of uncreatable numbers (c): 3115261
Final Density (c/n): 0.020768
Empirical Constant K = (c/n) * (log n)^2: 7.360819
Theoretical Constant K = 12 * C2: 7.921942
Difference (Theoretical - Empirical): 0.561123
Press any key to continue . . .