analyze-prime-numberslisted

# Analyze Prime Numbers Analyze prime numbers by selecting and applying the appropriate algorithm for the task at hand: primality testing, integer factorization, or prime distribution analysis. Verify results computationally and relate findings to the Prime Number Theorem. ## When to Use - Determining whether a given integer is prime or composite - Finding the complete prime factorization of an integer - Counting or listing primes up to a given bound - Verifying the Prime Number Theorem approximation for a specific range - Investigating properties of primes in a number-theoretic proof or computation ## Inputs - **Required**: The integer(s) to analyze, or a bound for distribution analysis - **Required**: Task type -- one of: primality test, factorization, or distribution analysis - **Optional**: Preferred algorithm (trial division, Miller-Rabin, Sieve of Eratosthenes, Pollard's rho) - **Optional**: Whether to produce a formal proof of primality or a computational verdict - **Optional**: Output format (factor tree, prime list, count, table) ## Procedure ### Step 1: Determine the Task Type Classify the request into one of three categories and select the appropriate algorithmic path. 1. **Primality test**: Given a single integer n, determine whether n is prime. 2. **Factorization**: Given a composite integer n, find its complete prime factorization. 3. **Distribution analysis**: Given a bound N, analyze the primes up to N (count, list, gaps, density). Record the task ty