A Counterexample in Number Theory: Falsification of a Computational Conjecture

We report the falsification of the following conjecture: For the sequence of primes of the form n^2+1, let p_k be the k-th such prime. The conjecture states that for all k >= 2, the gap between consecutive primes p_k and p_{k-1} satisfies: p_k - p_{k-1} < 2 * sqrt(p_k) * (ln(p_k))^0.8. This refines the gen. A counterexample was discovered computationally: witness = {'k': 11, 'p_k': 1297, 'p_prev': 677, 'gap': 620, 'bound': 348.18425797112735, 'base_n': 36}. This result was obtained by the SOVEREIGN autonomous research system.