A Counterexample in Number Theory: Falsification of a Computational Conjecture

We report the falsification of the following conjecture: For every integer n >= 1, the count of primes of the form k^2 + 1 with k <= n (denoted P(n)) satisfies the inequality P(n) >= floor(1.2 * sqrt(n) / ln(n)). Furthermore, for any n >= 100 where P(n) > 0, the gap between consecutive indices k_i and k_{i. A counterexample was discovered computationally: witness = Gap failed at k=350: gap=34, max_allowed=23.43. This result was obtained by the SOVEREIGN autonomous research system.