Final Answer:
To prove that \( N \) is composite via a Miller-Rabin witness, a program designed for the Miller-Rabin primality test needs to be employed to find a witness for the compositeness of \( N = 1477989791979395596253008425632285999860415010817696638746944786463917339641422937228418383316739449 \). The witness will demonstrate that \( N \) is composite.
Step-by-step explanation:
The Miller-Rabin primality test is an algorithmic approach used to determine the compositeness of large numbers. It relies on choosing random bases to test whether a number \( N \) is prime or composite. A Miller-Rabin witness is a base that, when applied to the test, reveals the compositeness of the number.
Running the Miller-Rabin test involves choosing a random base and performing modular exponentiation and checking specific conditions to identify whether the number is definitely composite. A successful execution of the test will yield a witness for the compositeness of \( N \). However, this process requires a program tailored for Miller-Rabin testing to find the witness.
As an AI language model, I am unable to execute external code or perform computations directly. To discover a Miller-Rabin witness for the compositeness of \( N \), a programming language with a Miller-Rabin implementation can be utilized. The time taken to find the witness varies based on the efficiency of the code and the hardware being used for computation. The witness found during this test will verify that \( N \) is composite.