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By Samuel Moy

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According to Archimedes, he who could use these relations to determine the unknown quantities could not be considered to be ignorant or incompetent in the realm of science, yet neither would he be worthy of being called a scholar. But he would reach perfection of knowledge if he succeeded in determining all these quantities by taking into account two additional conditions: the black and white bulls together can be arranged to form a square, while the brown and spotted bulls together can form a triangle with a single bull at its apex.

Then = (x + m)2 , with x2 + bx + c + h x2 + bx + c = x2 + 2mx + m2 , and so h x2 + bx + c = x (2m − b) + m2 − c . Squaring each side then yields h2 x2 + h2 bx + h2 c = x2 (2m − b)2 + 2x (2m − b) m2 − c + m2 − c 2 . We will obtain a rational x satisfying both equations by taking b±h . m= 2 This eliminates the x2 terms, leaving 2 m 2 − c − h2 c . x= 2 h b − 2 (2m − b) (m2 − c) Here, however, the sign of x will depend on the given values. (3) Ab¯ u K¯amil also treats the system x2 + bx = √ x2 + bx + h x2 + bx = .

Remark. • This problem shows that if a number can be written as a sum of two squares, then it can be written as such in infinitely many ways. 9), we will learn which numbers can be written as a sum of two squares. Example 3. 10 To find two square numbers having a given difference. Consider u2 − v 2 = k, with k given, for example (as Diophantus chooses) k = 60. ) We have, since this time x2 drops out, 2mx + m2 = k, and so k − m2 . 2m Each (positive) rational m with m2 < k yields a solution. With k = 60 and m = 3, Diophantus finds that x = 8 + 12 = v and u = 11 + 12 .

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