Red Planet Mars (1952)?
From this review:
A scientist picks up radio signals from Mars, which describe the utopian life there, including long lifespans and plentiful energy. When the signals start to deliver pronouncements from the Martian Supreme Leader, humanity realises that it is God speaking. There is mass panic.
However, all of this is a plot by an embittered ex-Nazi scientist working for the Communists who is bouncing the radio signals of the ionosphere to fool radio receivers. This backfires on the Communist government and they are overthrown
when they try to stop people worshipping.
When the Nazi is apprehended, an American scientist discovers a difference between his script and what the signals have been saying and realises that God is on Mars after all.
The line "How about Pi?" occurs at 22:24 in the video below:
The same review tackles the "political overtones" mentioned in the question, although apparently, someone didn't enjoy movie night:
The naivete of the film’s politics is incredible – it is not just the denouncement of the Soviet Union as evil but John Balderston’s tying up of religion in the mix as well.
In real life, the 1950s was a period of newfound religious fervour – Billy Graham began his massive charismatic Christian crusades. There was a strong sense of millenarism, of the end of the world being nigh due to the looming threats of the unleashed A-bomb and the growing Communist menace.
This is reflected in the film here where the Soviet Union is directly seen as a hotbed of atheism and religious suppression and by contrast the US is endorsed as an idyllic model of family, home and church-going Christians.
Main links blatantly taken from Boy asks 'How about pie?'