1. The problem is to create a mathematical formula that predicts the prime number sequence, starting from the 100th prime number. 2. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. 3. Unfortunately, there is no simple closed-form formula that generates prime numbers exactly. 4. However, the $n$th prime number, denoted $p_n$, can be approximated using the Prime Number Theorem: $$p_n \approx n \ln n$$ where $\ln n$ is the natural logarithm of $n$. 5. For better accuracy, a refined approximation is: $$p_n \approx n (\ln n + \ln \ln n - 1)$$ 6. Since the 100th prime is $p_{100}$, we can estimate it as: $$p_{100} \approx 100 (\ln 100 + \ln \ln 100 - 1)$$ 7. Calculating: $\ln 100 \approx 4.6052$ $\ln \ln 100 \approx \ln 4.6052 \approx 1.5272$ So, $$p_{100} \approx 100 (4.6052 + 1.5272 - 1) = 100 (5.1324) = 513.24$$ 8. The actual 100th prime is 541, so this approximation is close. 9. To generate primes starting from the 100th prime, one would typically use a prime sieve or primality test rather than a closed formula. 10. Summary: While no exact formula exists, the prime number theorem provides a good approximation for the $n$th prime number. Final answer: The approximate formula for the $n$th prime number starting at $n=100$ is $$p_n \approx n (\ln n + \ln \ln n - 1)$$