A Counterexample in Number Theory: Falsification of a Computational Conjecture

We report the falsification of the following conjecture: Let P_N be the set of primes of the form n^2+1 for 1 <= n <= N. Let A_N be the count of such primes where the generator n is itself a prime number. The conjecture states that for all N >= 1000, the ratio of the density of 'prime-generated' primes to . A counterexample was discovered computationally: witness = {'N': 50000, 'total_primes_form': 3613, 'prime_generated_count': 1, 'actual_ratio': 0.0002767783005812344, 'threshold': 0.09242333564642943}. This result was obtained by the SOVEREIGN autonomous research system.