Every prime number of the form 4n + 1 is the sum of two squares in exactly one way.
Replying to @AlgebraFact
I would argue that this is an arithmetic fact, rather than an algebra fact. And that it is also the tip of a vast iceberg, culminating (apparently) in the profoundly subtle, breathtakingly comprehensive regularities conjectured by Robert Langlands in 1967.
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