Abstract
This question is as old as number theory, and its solution is a classic in the field. The “hard” part of the solution is to see that every prime number of the form 4m + 1 is a sum of two squares. G. H. Hardy writes that this two square theorem of Fermat “is ranked, very justly, as one of the finest in arithmetic.” Nevertheless, our Book Proof below is recent and dates from 1990.
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References
I. Niven and H. S. Zuckerman:An Introduction to the Theory of Numbers, third edition, Wiley 1972.
M. Rubinstein and P. Sarnak: Chebyshev’s bias, Experimental Mathematics 3 (1994), 173–197.
S. Wagon: Editor’s corner: The Euclidean algorithm strikes again, Amer. Math. Monthly 77 (1990), 12. 5–129.
D. Zagier: A one-sentence proof that every prime p - 1 (mod 4) is a sum of two squares, Amer. Math. Monthly 77 (1990), 144.
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© 1998 Springer-Verlag Berlin Heidelberg
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Aigner, M., Ziegler, G.M. (1998). Representing numbers as sums of two squares. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22343-7_4
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DOI: https://doi.org/10.1007/978-3-662-22343-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-22345-1
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