TAS = EAS / square root of relative density
where relative density is actual density over ISA sea level density
MN = TAS / speed of sound at ISA sea level times square root of relative temperature
where relative temperature is actual temperature (in ēK) over ISA sea level temperature
Therefore:
MN = EAS / square root of relative density / standard speed of sound times square root of relative temperature
From the gas law equation:
relative pressure = relative density * relative temperature
so:
MN = EAS / standard speed of sound times square root of relative pressure
where relative pressure is actual pressure over ISA sea level pressure.
According to this, you only need pressure altitude and EAS (from undisturbed airstream dynamic pressure) to calculate MN. No need for temperature.