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Every Prime of the form 
is a sum of two Square Numbers in one unique way (up to the order of
Summands). The theorem was stated by Fermat 
, but the first published proof was by
Euler.
See also Sierpinski's Prime Sequence Theorem, Square Number
References
Hardy, G. H. and Wright, E. M. ``Some Notation.'' Th. 251 in
An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.