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MATHEMATICS MAGAZINE

Proof Without Words: Diophantus of Alexandria’s Sum of Squares Identity R O G E R B. N E L S E N

Lewis & Clark College Portland, OR [email protected]

Theorem [1]. If two positive integers are each sums of two squares, then their product is a sum of two squares in two different ways, i.e.,    2 (1) a + b2 c2 + d 2 = (ac + bd)2 + (ad − bc)2 and    2 a + b2 c2 + d 2 = (ad + bc)2 + (ac − bd)2    Example: 65 = 13 · 5 = 32 + 22 22 + 12 = 82 + 12 = 72 + 42 .

(2)

Proof. (of (1) when ad > bc, the case ad < bc is similar). ac+bd a 2+b 2

ad–bc

c 2+d 2

a 2

c2 +d

ad

ac

d2

2+ b c

bc

bd

 2  a 2 + b2 c2 + d 2 = (ac + bd)2 + (ad − bc)2 Exchanging c and d in the figure yields a proof of (2). REFERENCE 1. T. L. Heath, A History of Greek Mathematics Vol. II, Oxford Univ. Press, 1921. 481–482. Summary. theorem.

   We wordlessly prove a 2 + b2 c2 + d 2 = (ac ± bd)2 + (ad ∓ bc)2 using the Pythagorean

ROGER NELSEN (MR Author ID 237909) is a professor emeritus at Lewis & Clark College, where he taught mathematics and statistics for 40 years.

c Mathematical Association of America Math. Mag. 90 (2017) 134. doi:10.4169/math.mag.90.2.134.  MSC: Primary 11E25