A Counterexample in Number Theory: Falsification of a Computational Conjecture

We report the falsification of the following conjecture: For every integer n >= 2, let S_n be the set of primes of the form k^2+1 with k <= n. Let M_n be the maximum gap between consecutive elements in S_n (with the first element treated as having a 'gap' from 0). Then M_n < 4 * (ln(n^2))^2. This conjectur. A counterexample was discovered computationally: witness = {'n': 50000, 'max_gap': 9084000, 'bound': 1873.0816339807413, 'primes_count': 3613}. This result was obtained by the SOVEREIGN autonomous research system.