  The well known relation for ideal classical gas $\Delta \epsilon^2=kT^2 C_V$
which does not remain valid for quantum system is revisited. A new connection
is established between energy fluctuation and specific heat for quantum gases,
valid in the classical limit and the degenerate quantum regime as well. Most
importantly the proposed Biswas-Mitra-Bhattacharyya (BMB) conjecture (Biswas
$et.$ $al.$, J. Stat. Mech. P03013, 2015.) relating hump in energy fluctuation
and discontinuity of specific heat is proved and precised in this manuscript.
