In seeking to understand the structural necessity of infinite integers k not expressible as |6xy+x+y|, I developed a new tool to compare the slope of k yielding (1) a prime in 6k-1, (2) a prime in 6k+1, and (3) a twin prime of the form 6k+-1.
This module visually and empirically demonstrates how 6k-1 and 6k+1 have subtly different prime counts within them at a given k value; and how these interact in relation to the diminishing probability of k values which produce twin primes.
Remember, to understand this problem of the “Twin Prime Index Conjecture” we need to look at two separate ranges of integers. These are: (1) “k space” and (2) “n space”.
Both of these ranges cover all positive integers and so are infinite, but they are ultimately different domains.
Since we are working with non-zero k space primarily for the function n=|6k+1| to understand twin primes, the range of n space becomes defined as 1<=(k*6)+1. The relationship between primes in n space and twin prime indices in k space is demonstrated by the following sieve data.
| k Range: k>=10 | n Range: (k*6)+1 | No of Primes Less than or Equal to n in Range | No of Twin Prime Pairs (6k-1,6k+1) Less than or Equal to n in Range |
| 10 | 61 | 18 | 6 |
| 100 | 601 | 110 | 26 |
| 1,000 | 6,001 | 783 | 142 |
| 10,000 | 60,001 | 6,057 | 810 |
| 100,000 | 600,001 | 49,098 | 5,330 |
| 1,000,000 | 6,000,001 | 412,849 | 37,915 |
| 10,000,000 | 60,000,001 | 3,562,115 | 280,557 |
| 100,000,000 | 600,000,001 | 31,324,704 | 2,166,300 |
| 1,000,000,000 | 6,000,000,001 | 279,545,369 | 17,244,408 |
| 10,000,000,000 | 60,000,000,001 | 2,524,038,155 | 140,494,396 |
| 100,000,000,000 | 600,000,000,001 | 23,007,501,786 | 1,166,916,932 |
(Recall, we earlier demonstrated the equivalence of k \ |6xy+x+y| as equivalent to a Sieve of Eratosthenes adapted to extract twin prime indices from numbers of the form 6k+-1. So although the largest data in the table above is taken from a SoE, it is logically equivalent to our original formulation that k \ |6xy+x+y| are precisely the indices of a twin prime pair for non-zero x and y (which can be positive or negative). This allows us to test the hypothesis that a Diophantine equation like |6xy+x+y| cannot be surjective over infinitely many values of k using the SoE.)
The below module works on n space for convenience. Here is a data table for validation for the below output. Since we’re working with 6k-1,6k+1, just subtract 1 from the No of Twin Primes column, as we ignore 3,5.
| n range 0 to ? | No of Primes | No of Twin Primes |
| 10 | 4 | 2 |
| 100 | 25 | 8 |
| 1,000 | 168 | 35 |
| 10,000 | 1,229 | 205 |
| 100,000 | 9,592 | 1224 |
| 1,000,000 | 78,498 | 8169 |
| 10,000,000 | 664,579 | 58,980 |
| 100,000,000 | 5,761,455 | 440,312 |
| 1,000,000,000 | 50,847,534 | 3,424,506 |
| 10,000,000,000 | 455,052,511 | 27,412,679 |
| 100,000,000,000 | 4,118,054,813 | 224,376,048 |
Module Code (Python):
Note: Check out the Twin Prime Analysis Workbench as well. The following tool is not a module yet for the workbench but was built directly off of a Sieve of Eratosthenes using the same tools as the Workbench (numpy, pandas, matplotlib, numba, etc.).
def perform_pattern_analysis(is_prime_sieve: np.ndarray, n_limit: int, cores_to_use: int):
print("\n--- Prime Pattern Analysis (6k-1 vs 6k+1) ---")
k_max = n_limit // 6
if k_max < 20:
print("Not enough data for meaningful analysis.")
return
k_values = np.arange(1, k_max + 1)
print("\n[Commentary: This module analyzes the distribution of primes within two distinct")
print("sequences: those of the form 6k-1 and those of the form 6k+1. It tests for")
print("Chebyshev's Bias (the 'prime number race') and compares the density of these")
print("sequences to the density of the twin prime indices that are formed by their overlap.]")
primes_6k_minus_1_mask = is_prime_sieve[6 * k_values - 1]
primes_6k_plus_1_mask = is_prime_sieve[6 * k_values + 1]
# Get the actual lists of k's that satisfy each condition for accurate analysis
k_minus_1_list = k_values[primes_6k_minus_1_mask]
k_plus_1_list = k_values[primes_6k_plus_1_mask]
total_primes_in_seq = len(k_minus_1_list) + len(k_plus_1_list)
print(f"\nTotal Primes (>3 and <= {n_limit:,}): {total_primes_in_seq:,}")
print(f" - Primes of form 6k-1: {len(k_minus_1_list):,} ({len(k_minus_1_list) / total_primes_in_seq:.4%})")
print(f" - Primes of form 6k+1: {len(k_plus_1_list):,} ({len(k_plus_1_list) / total_primes_in_seq:.4%})")
twin_prime_k_mask = primes_6k_minus_1_mask & primes_6k_plus_1_mask
k_twin_list = k_values[twin_prime_k_mask]
print(f"\nValidation: Found {len(k_twin_list):,} twin prime pairs of form 6k+/-1.")
def analyze_sequence(k_list, name):
"""Analyzes a specific list of k indices."""
if len(k_list) < 20:
print(f"\nAnalysis for {name}: Not enough data points.")
return None, None
c = np.arange(1, len(k_list) + 1)
density = c / k_list
print(f"\nAnalysis for {name}: Final Density = {density[-1]:.6f}")
# Perform regression on the actual data points for this sequence
X = (1 / np.log(6 * k_list[10:])).reshape(-1, 1)
y = density[10:]
model = LinearRegression().fit(X, y)
print(f" Linear Fit R-squared = {r2_score(y, model.predict(X)):.6f}")
return (k_list, density), model
print("\n--- Density and Regression Analysis ---")
data_minus_1, model_minus_1 = analyze_sequence(k_minus_1_list, "k Yielding 6k-1 Primes")
data_plus_1, model_plus_1 = analyze_sequence(k_plus_1_list, "k Yielding 6k+1 Primes")
data_twin_k, model_twin_k = analyze_sequence(k_twin_list, "k Yielding Twin Prime")
if input("Generate comparison plot? (Y/N): ").strip().lower() == 'y':
plt.figure(figsize=(14, 8)); plt.style.use('seaborn-v0_8-whitegrid')
def plot_data(data, model, color, label):
"""Helper function to correctly plot each sequence and its true fit."""
if data is None or model is None: return
k_sequence, density = data
sample_step = max(1, len(k_sequence) // 2000)
plt.scatter(k_sequence[::sample_step], density[::sample_step], s=10, alpha=0.5, label=f'{label} Density', color=color)
# Generate the fit line based on the actual range of this sequence's k values
plot_k_values = np.geomspace(k_sequence[10], k_sequence[-1], 2000)
fit_x_transform = 1 / np.log(6 * plot_k_values)
plt.plot(plot_k_values, model.predict(fit_x_transform.reshape(-1, 1)), color=color, label=f'{label} Fit')
plot_data(data_minus_1, model_minus_1, 'C0', 'k Yielding 6k-1 Primes')
plot_data(data_plus_1, model_plus_1, 'C1', 'k Yielding 6k+1 Primes')
plot_data(data_twin_k, model_twin_k, 'C2', 'k Yielding Twin Prime')
plt.xscale('log'); plt.title('Comparative Density of Prime and Twin Prime Index Sequences'); plt.xlabel('k')
plt.ylabel('Cumulative Density (Count(k) / k)'); plt.legend(); plt.grid(True, which="both", ls="--"); plt.show()Findings:
Confirmation of Chebyshev’s Bias: Across all tested scales, from k=10³ to k=10¹⁰, a subtle but consistent bias is observed where primes of the form 6k-1 are more numerous than primes of the form 6k+1. While the absolute lead of the 6k-1 sequence grows, the proportional difference narrows, with the distribution approaching a 50/50 split as k becomes very large. The visual representation of this is the small but unwavering gap between the blue and orange density curves.
At each range of the analysis, the clear gap in the slopes of k yielding prime in 6k-1 and 6k+1 is apparent.
For k=10^3
For k=10^9
Divergent Trajectory of Twin Primes: The most significant finding is the dramatic divergence in the density of twin prime indices compared to the individual prime sequences. While the densities of 6k-1 and 6k+1 primes follow closely related downward curves, the density of twin prime indices (k where both are prime) falls off at a much faster rate. This visually represents the compounding improbability of meeting two prime conditions simultaneously. The ever-widening gap between the upper curves and the lower green curve illustrates that a twin prime is a significantly rarer event than a prime in either individual sequence.
Stability and Predictability of Trends: The linear regression models for all three sequences demonstrate exceptionally high R-squared values (often exceeding 0.999 for the individual prime sequences and >0.96 for the twin prime sequence). This indicates that the observed density curves are not erratic but are highly stable and predictable, adhering closely to a logarithmic trend.
The R-squared of the twin prime sequence is necessarily less strong, because it has far fewer data points to plot against compared to prime yielding k for 6k-1 and prime yielding k for 6k+1.
Conclusion and Relation to the Diophantine Hypothesis:
These empirical results are crucial for the core hypothesis. The fact that the twin prime index density, while diminishing rapidly, establishes a stable, non-zero, and predictable downward curve strongly suggests it will never reach zero.
This aligns perfectly with the conjecture that the Diophantine set k = |6xy+x+y| has an infinite complement. The persistent, non-terminating green curve is the graphical signature of this infinite set of “uncreatable” k values, each corresponding to a twin prime pair.
Outputs:
[n range = 1,000]
--- Prime Pattern Analysis (6k-1 vs 6k+1) ---
[Commentary: This module analyzes the distribution of primes within two distinct
sequences: those of the form 6k-1 and those of the form 6k+1. It tests for
Chebyshev's Bias (the 'prime number race') and compares the density of these
sequences to the density of the twin prime indices that are formed by their overlap.]
Total Primes (>3 and <= 1,000): 166
- Primes of form 6k-1: 86 (51.8072%)
- Primes of form 6k+1: 80 (48.1928%)
Validation: Found 34 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.524390
Linear Fit R-squared = 0.979390
Analysis for k Yielding 6k+1 Primes: Final Density = 0.481928
Linear Fit R-squared = 0.974979
Analysis for k Yielding Twin Prime: Final Density = 0.231293
Linear Fit R-squared = 0.971335
Generate comparison plot? (Y/N): y
[n range = 10,000]
Total Primes (>3 and <= 10,000): 1,227
- Primes of form 6k-1: 616 (50.2037%)
- Primes of form 6k+1: 611 (49.7963%)
Validation: Found 204 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.371756
Linear Fit R-squared = 0.995786
Analysis for k Yielding 6k+1 Primes: Final Density = 0.367629
Linear Fit R-squared = 0.994496
Analysis for k Yielding Twin Prime: Final Density = 0.123263
Linear Fit R-squared = 0.967063
Generate comparison plot? (Y/N): y
[n range = 100,000]
Total Primes (>3 and <= 100,000): 9,590
- Primes of form 6k-1: 4,806 (50.1147%)
- Primes of form 6k+1: 4,784 (49.8853%)
Validation: Found 1,223 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.288389
Linear Fit R-squared = 0.998738
Analysis for k Yielding 6k+1 Primes: Final Density = 0.287069
Linear Fit R-squared = 0.997805
Analysis for k Yielding Twin Prime: Final Density = 0.073387
Linear Fit R-squared = 0.973823
Generate comparison plot? (Y/N): y
[n range = 1,000,000]
Total Primes (>3 and <= 1,000,000): 78,496
- Primes of form 6k-1: 39,265 (50.0217%)
- Primes of form 6k+1: 39,231 (49.9783%)
Validation: Found 8,168 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.235594
Linear Fit R-squared = 0.999135
Analysis for k Yielding 6k+1 Primes: Final Density = 0.235391
Linear Fit R-squared = 0.999257
Analysis for k Yielding Twin Prime: Final Density = 0.049010
Linear Fit R-squared = 0.958276
Generate comparison plot? (Y/N): y
[n range = 10,000,000]
Total Primes (>3 and <= 10,000,000): 664,577
- Primes of form 6k-1: 332,383 (50.0142%)
- Primes of form 6k+1: 332,194 (49.9858%)
Validation: Found 58,979 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.199430
Linear Fit R-squared = 0.999147
Analysis for k Yielding 6k+1 Primes: Final Density = 0.199317
Linear Fit R-squared = 0.999786
Analysis for k Yielding Twin Prime: Final Density = 0.035387
Linear Fit R-squared = 0.944281
[n range = 100,000,000]
Total Primes (>3 and <= 100,000,000): 5,761,453
- Primes of form 6k-1: 2,880,936 (50.0036%)
- Primes of form 6k+1: 2,880,517 (49.9964%)
Validation: Found 440,311 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.172856
Linear Fit R-squared = 0.999405
Analysis for k Yielding 6k+1 Primes: Final Density = 0.172831
Linear Fit R-squared = 0.999869
Analysis for k Yielding Twin Prime: Final Density = 0.026419
Linear Fit R-squared = 0.951757
[n range = 1,000,000,000]
Total Primes (>3 and <= 1,000,000,000): 50,847,532
- Primes of form 6k-1: 25,424,819 (50.0021%)
- Primes of form 6k+1: 25,422,713 (49.9979%)
Validation: Found 3,424,505 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.152549
Linear Fit R-squared = 0.999685
Analysis for k Yielding 6k+1 Primes: Final Density = 0.152536
Linear Fit R-squared = 0.999885
Analysis for k Yielding Twin Prime: Final Density = 0.020547
Linear Fit R-squared = 0.964602
Generate comparison plot? (Y/N): y
[n range = 10,000,000,000]
Total Primes (>3 and <= 10,000,000,000): 455,052,509
- Primes of form 6k-1: 227,529,386 (50.0007%)
- Primes of form 6k+1: 227,523,123 (49.9993%)
Validation: Found 27,412,678 twin prime pairs of form 6k+/-1.
--- Density and Regression Analysis ---
Analysis for k Yielding 6k-1 Primes: Final Density = 0.136518
Linear Fit R-squared = 0.999851
Analysis for k Yielding 6k+1 Primes: Final Density = 0.136514
Linear Fit R-squared = 0.999916
Analysis for k Yielding Twin Prime: Final Density = 0.016448
Linear Fit R-squared = 0.978129
Generate comparison plot? (Y/N): y