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

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Description: b-d Please! 6.16 Some Number Theory in the "Arithmetica"(a) Establish the identities(a² + b?)(c² + d?) = (ac ± bd)2 + (ad + bc)²and use them to express 481 = (13)(37) as the sum of 2 squares in 2different ways.%3DThese identities were given later, in

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Chapter2: Second-order Linear Odes

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b-d Please!

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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