On the largest size of (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 ) -core partitions is true. We also show that the number of ( t , t + 1 , t + 2 ) -core partitions with the largest size is 1 or 2 based on the parity of t . More generally, the largest size of ( t , t + 1 , ? , t + p ) -core partitions and the number of such partitions with the largest size are determined.