  We conduct a rigorous investigation into the thermodynamic instability of
ideal Bose gas confined in a cubic box, without assuming thermodynamic limit
nor continuous approximation. Based on the exact expression of canonical
partition function, we perform numerical computations up to the number of
particles one million. We report that if the number of particles is equal to or
greater than a certain critical value, which turns out to be 7616, the ideal
Bose gas subject to Dirichlet boundary condition reveals a thermodynamic
instability. Accordingly we demonstrate - for the first time - that, a system
consisting of finite number of particles can exhibit a discontinuous phase
transition featuring a genuine mathematical singularity, provided we keep not
volume but pressure constant. The specific number, 7616 can be regarded as a
characteristic number of 'cube' that is the geometric shape of the box.
