On the largest size of (t,t+1,…,t+p)(t,t+1,…,t+p)-core partitions

In this paper we prove that Amdeberhan’s conjecture on the largest size of (t,t+1,t+2)(t,t+1,t+2)-core partitions is true. We also show that the number of (t,t+1,t+2)(t,t+1,t+2)-core partitions with the largest size is 1 or 2 based on the parity of tt. More generally, the largest size of (t,t+1,…,t+p)(t,t+1,…,t+p)-core partitions and the number of such partitions with the largest size are determined.