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6.16 Some Number Theory in the "Arithmetica" (a) Establish the identities (a² + b?)(c² + d?) = (ac ± bd)2 + (ad + bc)² and use them to express 481 = (13)(37) as the sum of 2 squares in 2 different ways. %3D These identities were given later, in 1202, by Fibonacci in his Liber abaci. They show that the product of 2 numbers, each expressible as the sum of 2 squares, is also expressible as the sum of 2 squares. It can be shown that these identities include the addition formulas for the sine and cosine. The identities later became the germ of the Gaussian theory of arithmetical quadratic forms and of certain developments in modern algebra. (b) Express 1105 = (5)(13)(17) as the sum of 2 squares in 4 different ways. In the following 2 problems, “number" means "positive rational number." %3D (c) If m and n are numbers differing by 1, and if x, y, a are numbers such that x + a = m², y + a = n2, show that xy + a is a square number. (d) If m is any number and x = m2, y = (m + 1)², z = 2(x + y + 1), show that the 6 numbers xy + x + y, yz + y + z, zx + z + x, xy + z, yz + x, zx + y are all square numbers. %3D %3D %3D

6.16 Some Number Theory in the "Arithmetica" (a) Establish the identities (a² + b?)(c² + d?) = (ac ± bd)2 + (ad + bc)² and use them to express 481 = (13)(37) as the sum of 2 squares in 2 different ways. %3D These identities were given later, in 1202, by Fibonacci in his Liber abaci. They show that the product of 2 numbers, each expressible as the sum of 2 squares, is also expressible as the sum of 2 squares. It can be shown that these identities include the addition formulas for the sine and cosine. The identities later became the germ of the Gaussian theory of arithmetical quadratic forms and of certain developments in modern algebra. (b) Express 1105 = (5)(13)(17) as the sum of 2 squares in 4 different ways. In the following 2 problems, “number" means "positive rational number." %3D (c) If m and n are numbers differing by 1, and if x, y, a are numbers such that x + a = m², y + a = n2, show that xy + a is a square number. (d) If m is any number and x = m2, y = (m + 1)², z = 2(x + y + 1), show that the 6 numbers xy + x + y, yz + y + z, zx + z + x, xy + z, yz + x, zx + y are all square numbers. %3D %3D %3D

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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6.16 Some Number Theory in the "Arithmetica" (a) Establish the identities (a² + b?)(c² + d?) = (ac ± bd)2 + (ad + bc)² and use them to express 481 = (13)(37) as the sum of 2 squares in 2 different ways. %3D These identities were given later, in 1202, by Fibonacci in his Liber abaci. They show that the product of 2 numbers, each expressible as the sum of 2 squares, is also expressible as the sum of 2 squares. It can be shown that these identities include the addition formulas for the sine and cosine. The identities later became the germ of the Gaussian theory of arithmetical quadratic forms and of certain developments in modern algebra. (b) Express 1105 = (5)(13)(17) as the sum of 2 squares in 4 different ways. In the following 2 problems, “number" means "positive rational number." %3D (c) If m and n are numbers differing by 1, and if x, y, a are numbers such that x + a = m², y + a = n2, show that xy + a is a square number. (d) If m is any number and x = m2, y = (m + 1)², z = 2(x + y + 1), show that the 6 numbers xy + x + y, yz + y + z, zx + z + x, xy + z, yz + x, zx + y are all square numbers. %3D %3D %3D
Transcribed Image Text:6.16 Some Number Theory in the "Arithmetica" (a) Establish the identities (a² + b?)(c² + d?) = (ac ± bd)2 + (ad + bc)² and use them to express 481 = (13)(37) as the sum of 2 squares in 2 different ways. %3D These identities were given later, in 1202, by Fibonacci in his Liber abaci. They show that the product of 2 numbers, each expressible as the sum of 2 squares, is also expressible as the sum of 2 squares. It can be shown that these identities include the addition formulas for the sine and cosine. The identities later became the germ of the Gaussian theory of arithmetical quadratic forms and of certain developments in modern algebra. (b) Express 1105 = (5)(13)(17) as the sum of 2 squares in 4 different ways. In the following 2 problems, “number" means "positive rational number." %3D (c) If m and n are numbers differing by 1, and if x, y, a are numbers such that x + a = m², y + a = n2, show that xy + a is a square number. (d) If m is any number and x = m2, y = (m + 1)², z = 2(x + y + 1), show that the 6 numbers xy + x + y, yz + y + z, zx + z + x, xy + z, yz + x, zx + y are all square numbers. %3D %3D %3D
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