The 4th Landau problem states that there are infinitely many prime numbers of the form n^2+1, where n is a positive integer. Current partial results show that for infinitely many n, the largest prime factor of n^2+1 exceeds n^c for some exponent c<2.

Currently, the best unconditional result shows that infinitely many values of n^2+1 have a prime factor larger than n^1.312 (https://arxiv.org/abs/2505.00493). This market asks: By Dec 31, 2050, what will be the largest proven exponent c such that infinitely many n satisfy P(n^2+1)>n^c?

Resolution will be based on published peer-reviewed proofs or widely accepted preprints (arXiv or equivalent). Only unconditional results count — conditional results (e.g. assuming GRH) do not resolve the market unless the assumption is also proved before 2050. If multiple results exist, the highest valid exponent among the listed options will be chosen.

If a full proof is found showing that infinitely many primes of the form n^2+1 exist, the market will resolve to 2 (solved).