Question

Diophantus passed à of his life in childhood, in youth, and more as a bachelor. Five years after his marriage was born a son who died 4 before his father, at his father's [final] age."How old was years Diophantus when he died? (b) Solve the following problem, which appears in Diophantus' Arithmetica (Problem 17, Book I): Find 4 numbers, the sum of every arrangement 3 at a time being given; say, 22, 24, 27, and 20 (c) Solve the following problem, also found in the Arithmetica (Problem 16, Book VI): In the right triangle ABC, right angled at C, AD bisects angle A. Find the set of smallest integers for AB, AD, AC, BD, DC such that DC:CA:AD = 3:4:5 (d) Augustus De Morgan, who lived in the nineteenth century, proposed the conundrum: "I was x years old in the year x2." When was he born? 6.16 Some Number Theory in the "Arithmetica" (a) Establish the identities (a2 + b2)(c2 + d2)-(ac bd)2 + (ad bc)2 and use them to express 481 = (13)(37) as the sum of 2 squares in 2 different ways 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." that x + a that the 6 numbers xy +x+y, yzt y t z, zx (c) If m and n are numbers differing by 1, and if x, y, a are numbers such m2,y + a (d) If m is any number and x -m2, y (m n2, show that xy + a is a square number. 1)2, z- 2(x y1), show z+ x, xy + z, yz + x, zx + y are all square numbers 6.17 Problems from Pappus (a) In Book III of Pappus' Mathematical Collection, we find the following interesting geometrical representation of some means. Take B on seg- ment AC, B not being the midpoint O of AC. Erect the perpendicular to AC at B to cut the semicircle on AC in D, and let F be the foot of the perpendicular from B on OD. Show that OD, BD, FD represent the

i cant seem to figure out 6.16

Answer 1

a) (a^2 + b^2)(c^2 + d^2) = (ac plusminus bd)^2 + (ad - be)^2 R.H.S: - (ac plusminus bd)^2 + (ad plusminus bc)^2 = a^2 c^2 plusminus 2abcd + b^2d^2 + a^2 d^2 plusminus 2abcd + b^2c^2 = a^2 (c^2 + d^2) + b^2 (c^2 + d^2) = (a^2 + b^2) (c^2 + d^2) = L.H.S < proved > Now, 481 = (13)(37) if a^2 + b^2 = 13, c^2 + d^2 = 37 then, a^2 = 9 b^2 = 4 c^2 = 36 d^2 = 1 implies a = 3, b = 2 c = 6, d = 1 Now, (ac + bd)^2 = (18 + 2)^2 = (20)^2 = 400 105 = (13 times 5) (17) = (65) (17) a^2 + b^2 = 65 a = 8, b = 1 c^2 + d^2 = 17 c = 4, d = 9 (ac plusminus bd)^2 = (32 plusminus 1)^2 = 1089 on 961 (ad plusminus bc)^2 = (8 plusminus 4)^2 = 16 or