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 1202, by Fibonacci in his Liberabaci. They show that the product of 2 numbers, each expressible asthe sum of 2 squares, is also expressible as the sum of 2 squares. It canbe shown that these identities include the addition formulas for the sineand cosine. The identities later became the germ of the Gaussian theoryof arithmetical quadratic forms and of certain developments in modernalgebra.(b) Express 1105 = (5)(13)(17) as the sum of 2 squares in 4 different ways.In the following 2 problems, “number" means "positive rationalnumber."%3D(c) If m and n are numbers differing by 1, and if x, y, a are numbers suchthat 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), showthat 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

Name: b-d Please! 6.16 Some Number Theory in the "Arithmetica"(a) Establish
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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