Prime Time Redux

CONTINUED FROM - Strong Nuclear, Weak Nuclear, Nuclear-Nuclear, and Light, Why Not? - #5 by Josef_Founder

Static:

import math

# Step 1: Define the dynamic adjusted prime formula
def adjusted_prime_formula(n, alpha=1, decay_constant=2, r_scale=1, nu=1):
    """
    Calculate a potential prime based on the refined framework formula:
    p_n = (alpha * (F_n + nu)) / r_n * (1 - epsilon_n)
    
    Args:
    n (int): The prime number position to estimate.
    alpha (float): Scaling factor for force calculation.
    decay_constant (float): Exponent to modulate the decay effect of the error term.
    r_scale (float): Scaling factor for distance parameter.
    nu (float): Harmonic scaling factor to adjust for the force.
    
    Returns:
    float: Calculated potential prime number based on the formula.
    """
    # Adjust force and distance dynamically based on n
    F_n = n**2  # A polynomial growth of force for large numbers (n^2 for scaling)
    r_n = math.log(n)  # Logarithmic scale for distance factor
    
    # Calculate error term epsilon_n based on the adjusted parameters
    epsilon_n = (F_n / n) ** (1 / 2) * (1 / math.log(n))
    
    # Calculate the prime estimate with the formula
    prime_estimate = (alpha * (F_n + nu)) / r_n * (1 - epsilon_n)
    
    return prime_estimate

# Step 2: Simple primality check
def is_prime(n):
    """
    Check if a number is prime using a simple primality test.
    
    Args:
    n (int): The number to check for primality.
    
    Returns:
    bool: True if the number is prime, False otherwise.
    """
    if n <= 1:
        return False
    for i in range(2, int(math.sqrt(n)) + 1):
        if n % i == 0:
            return False
    return True

# Step 3: Generate primes using the adjusted formula and check them
def generate_primes(limit):
    """
    Generate and validate primes using the adjusted prime formula.
    
    Args:
    limit (int): The upper limit for generating primes.
    
    Returns:
    list: A list of prime numbers identified through the formula.
    """
    primes = []
    for n in range(2, limit + 1):
        prime_value = adjusted_prime_formula(n)
        if is_prime(int(prime_value)):
            primes.append(int(prime_value))
    return primes

# Step 4: Test the framework with larger primes
limit = 1000  # Change this to a larger number for bigger primes
primes = generate_primes(limit)

# Output the prime numbers found
print("Primes found using adjusted framework:", primes)

Dynamic:

import math

# Step 1: Define the dynamic adjusted prime formula with automatic parameter selection
def adjusted_prime_formula(n):
    """
    Calculate a potential prime based on dynamically adjusted force, distance, and error term.
    
    Args:
    n (int): The prime number position to estimate.
    
    Returns:
    float: Calculated potential prime number based on the formula.
    """
    # Step 1a: Automatically select parameters based on n
    alpha = n / 10  # Scaling factor for force calculation (dynamically adjusted)
    decay_constant = 2 + math.log(n)  # Decay constant modulated by log(n)
    r_scale = n ** (1 / 3)  # Distance scaling for larger n (cube root scaling)
    nu = math.sin(n)  # Harmonic scaling factor based on n
    
    # Step 1b: Adjust force and distance dynamically based on n
    F_n = n**2  # A polynomial growth of force for large numbers (n^2 for scaling)
    r_n = math.log(n)  # Logarithmic scale for distance factor
    
    # Step 1c: Calculate error term epsilon_n based on the adjusted parameters
    epsilon_n = (F_n / n) ** (1 / 2) * (1 / math.log(n))
    
    # Step 1d: Calculate the prime estimate with the formula
    prime_estimate = (alpha * (F_n + nu)) / r_n * (1 - epsilon_n)
    
    return prime_estimate

# Step 2: Simple primality check
def is_prime(n):
    """
    Check if a number is prime using a simple primality test.
    
    Args:
    n (int): The number to check for primality.
    
    Returns:
    bool: True if the number is prime, False otherwise.
    """
    if n <= 1:
        return False
    for i in range(2, int(math.sqrt(n)) + 1):
        if n % i == 0:
            return False
    return True

# Step 3: Generate primes using the adjusted formula and check them
def generate_primes(limit):
    """
    Generate and validate primes using the adjusted prime formula.
    
    Args:
    limit (int): The upper limit for generating primes.
    
    Returns:
    list: A list of prime numbers identified through the formula.
    """
    primes = []
    for n in range(2, limit + 1):
        prime_value = adjusted_prime_formula(n)
        if is_prime(int(prime_value)):
            primes.append(int(prime_value))
    return primes

# Step 4: Test the framework with larger primes
limit = 1000  # Change this to a larger number for bigger primes
primes = generate_primes(limit)

# Output the prime numbers found
print("Primes found using adjusted framework:", primes)




PROMPT:
Test and adapt our framework if applicable.






“The current framework works for smaller primes, but further testing and refinement are needed for large primes. The self-checking modular approach combined with harmonic adjustment ensures the validity of primes generated, but optimizing the equation to handle larger values without computational overhead will require iterative adjustments.”

PROMPT:
“Use the self-checking of our framework to solve for higher prime numbers and to test.”







PROMPT:
Extend this approach to larger and larger prime numbers adjusting our framework as necessary to complete the framework for any given context or scope.







import math

# Step 1: Define the error adjustment and scaling functions

def error_term(Fn, pn):
    """Calculate the error term for a given prime number."""
    return (Fn / pn) ** (1 / 2) * (1 / math.log(pn))

def force_param(n, k=1.1):
    """Calculate the force parameter, scaling with the nth iteration."""
    return n ** k

def distance_param(n):
    """Calculate the distance parameter using logarithmic scaling."""
    return math.log(n)

# Step 2: Generalized formula for the nth prime number

def calc_prime(n, alpha=1, nu=1):
    """Calculate the nth prime number using the generalized formula."""
    Fn = force_param(n)
    rn = distance_param(n)
    epsilon_n = error_term(Fn, n)
    
    # Calculate the prime approximation
    pn = (alpha * (Fn + nu)) / rn * (1 - epsilon_n)
    
    return pn

# Step 3: Prime testing function with dynamic scaling and error correction
def test_prime(n):
    """Test if a number is prime using the framework."""
    pn = calc_prime(n)
    
    # Round to the nearest integer to simulate a prime number
    rounded_pn = round(pn)
    
    # Checking if the rounded value is actually prime
    if is_prime(rounded_pn):
        print(f"Prime {n} calculated: {rounded_pn}")
    else:
        print(f"Calculated value {rounded_pn} is not prime. Adjusting error term...")
        # Try adjusting the error term and recalculate
        epsilon_n = error_term(force_param(n), n)
        pn_adjusted = (alpha * (force_param(n) + nu)) / distance_param(n) * (1 - epsilon_n)
        print(f"Adjusted prime for n={n}: {round(pn_adjusted)}")

# Step 4: Simple prime checking function for validation
def is_prime(n):
    """Check if a number is prime."""
    if n <= 1:
        return False
    for i in range(2, int(math.sqrt(n)) + 1):
        if n % i == 0:
            return False
    return True

# Test for primes starting from n=1 upwards
for n in range(1, 15):  # Test first 15 primes
    test_prime(n)

BRANCH1 - Working on an Alternative Approach?

PROMPT:
Analyze the decay of cancelling terms into the infinite and adjust our framework accordingly






PROMPT:
Compare our framework to the traditional method of solving for prime








PROMPT:
Compare them mathematically (I had to ask twice because silly bot reverts to thinking m = mass, not meters)







BRANCH2 - Working on an Alternative Approach?

PROMPT:
We still lack dynamically selecting parameters and auto-selecting error correction.