On the largest sizes of certain simultaneous core partitions with distinct parts

We derive the largest sizes of (t,mt+1) and (t,mt−1)-core partitions with distinct parts, and prove that the numbers of such partitions with the largest sizes are at most 2. This work is motivated by Amdeberhan's conjecture on (t,t+1)-core partitions with distinct parts and subsequent research on the enumeration, largest sizes and the average sizes of simultaneous core partitions with distinct parts.