  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.
