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Ch. 3.3 Finding Prime Number Greater or Lesser Than a Given Value

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Program 1

Finding Prime Number Greater or Lesser Than a Given Value

Python 3.11 Ready 

                                

                            


                            
                            

                                

                                    
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 Mathematical Problem Formulation

                                        
        


    
Input

    
Given number (e.g., 53).


    
Call next_prime(given_value)

    
Increment num by 1.


    
Check if num is Prime:

    
Start loop: Check divisibility from 2 to num - 1.

    
    
        
  • If num is divisible by any i, return False.
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  • If no divisors are found, return True.
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  • If not prime, increment num and repeat the check.
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  • Once a prime is found, return that prime number.
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Store the result in next_prime_greater.


    
Call previous_prime(given_value)

    
Decrement num by 1.


    
Check if num is Prime:

    
Start loop: Check divisibility from 2 to num - 1.

    
    
        
  • If num is divisible by any i, return False.
    • 
        
  • If no divisors are found, return True.
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  • If not prime, decrement num and repeat the check.
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  • Once a prime is found, return that prime number.
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Store the result in previous_prime_lesser.

                                    

                                    
                                                                    

                            

                                                    

                    

                

                                

                    
                    

                        

                            Program 2
                            
Finding Prime Number Greater or Lesser Than a Given Value

                        

                        

                            
                            
                            
                        

                    


                    
                    

                        
                        

                            

                                 Python 3.11
                                Ready
                            

                            

                                
                                
                            

                            
                            

                                

                                    
                                    
                                    
                                    
                                

                                

                                    
                                

                            

                        


                        
                        

                            

                                
                                
                                                            


                            
                            

                                

                                

                            


                            
                            

                                

                                    
Click "Run Code" to execute script and render figures.

                                

                            


                            
                                                    

                    

                

                            
            
            
            
            
            

                

                    

                         Theoretical Principles & Computational Physics Guide
                    

                    

                        Prime Search Near a Threshold: Next & Previous Prime Algorithms                    

                    

                         Dr. Alorika Chatterjee & Dr. Anirban Shaw
                         Science & Physics Learners
                         Python 3 / NumPy / SciPy
                    

                


                
                

                    

                         1. Core Principles & Theoretical Framework
                    

                    

                        
                    

                        
 Threshold Seeking: Finds the immediate next prime $p > N$ or greatest prime $p < N$ relative to an arbitrary scientific threshold.

                        
 Prime Gap Bounds: Bertrand's postulate guarantees at least one prime exists between $N$ and $2N - 2$.

                        
 Cryptographic & Hashing Roles: Used in physics simulations for pseudo-random number generator modulus selection.

                    
                    

                


                
                

                    

                         2. Mathematical Formulation & Algorithmic Pipeline
                    

                    

                        
                    
Prime gap $\Delta_p$ and Bertrand's Postulate guarantee:

                    
$$ \forall N > 1, \quad \exists p \in \text{Primes} \quad \text{such that} \quad N < p < 2N - 2 $$
                    

                


                
                

                    

                         3. Numerical Mechanics & Implementation Logic
                    

                    

                        
                    
    
                        
  • Search Direction: To find next prime, set candidate $c = N + 1$ (or $N + 2$ if $N$ is odd); increment by 2.
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  • Primality Sub-Check: Apply the $\mathcal{O}(\sqrt{c})$ primality test to each candidate.
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  • Termination: Break and return the first candidate satisfying primality criteria.
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                         4. Relevant Computational Practice Exercises
                    

                    

                                                    

                                Exercise 1
                                

                                    Find the nearest prime both above and below Avogadro's truncated integer representation $602214$.                                

                                
                            

                                                    

                                Exercise 2
                                

                                    Compute and plot prime gaps $\Delta_n = p_{n+1} - p_n$ for the first 1000 primes above $10^4$.                                

                                
                            

                                                    

                                Exercise 3
                                

                                    Verify Bertrand's Postulate computationally for 50 arbitrary values of $N \in [10^2, 10^5]$.                                

                                
                            

                                            

                


                
                

                     Standard Literature & References:
                    
    
                                                    
  • G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers", Oxford University Press (2008).
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