History of Mathematics Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: April 7, 2014 Student: Spring 2014
Problem A1. Given a square ABCD, equilateral triangles ABX and BCY are constructed as shown in the diagram. Show that D, X, Y are collinear. D X C Y A B
Problem A2. In the diagram, BAC and PDQ are parallel lines, and AP, AQ are bisectors of angles BAD and CAD respectively. Prove that PD = DQ. B P A D Q C
Problem A3. ABCD is a parallelogram; X and Y are the midpoints of the sides AB and CD respectively. Prove that CX and AY trisect the diagonal BD. D Y C A X B
Problem A4. Calculate the angle θ in the regular pentagram. θ
Problem A5. ABC is an isosceles triangle with B = C =80. D and E are points on AC and AB respectively such that DBC =60 and ECB =50. Calculate BDE. Hint: Let F be the point on AC such that BF = BC. Show that DF = EF = BF. A D E B 60 50 C
Problem A6. Each angle of regular m-gon is 3 of an angle of a regular 4 n-gon. Find all possibilities of (m, n).
Problem A7. ABCD is a square inscribed in a semicircle with diameter XY. Show that B divides AY in the golden ratio. D C X A O B Y
Problem A8. Three equal segments A 1 B 1, A 2 B 2, A 3 B 3 are positioned in such a way that the endpoints B 2, B 3 are the midpoints of A 1 B 1, A 2 B 2 respectively, while the endpoints A 1, A 2, A 3 are on a line perpendicular to A 1 B 1. Calculate the ratio A 1A 3 A 1 A 2. B 2 B 1 B 3 A 1 A 2 A 3
Problem A9. Given an equilateral triangle ABC, erect a square BCDE externally on the side BC. Construct the circle, center C, passing through E, to intersect the line AB at F. Then, AB divides AF in the golden ratio. C D E A B F
Problem A10. Let D and E be the midpoints of the sides AB and AC of an equilateral triangle ABC. If the line DE intersects the circumcircle of ABC at F, calculate the ratio DE : EF. D A O E F B C
Problem B1. Reduce the fraction 6527 5429 write it as a finite continued fraction. to its lowest terms, and also
Problem B2. Prove that 3 cannot be written in the form p q common divisors. for integers p and q without
Problem B3. Find the continued fraction expansion of 3, exhibiting its periodicity.
Problem B4. Let σ(n) be the sum of divisor function. Show that (a) q is a prime number if and only if σ(q) =1+q; (b) σ(2 k 1 )=2 k 1.
Problem B5. A number σ(n) is called abundant if σ(n) > 2n and deficient if σ(n) < 2n. (a) What is the smallest even abundant number? (b) Show that 945 is an abundant number? 1 1 It is the smallest odd abundant number.
Problem B6. Determine if M 13 = 8191 and M 23 = 8388607 are prime numbers.
Problem B7. (Lucas - Lehmer test for Mersenne primes) Start with v 2 =4. Form a sequence v 3,v 4,...,v k by setting v i+1 = vi 2 2. Assume k prime. M k is prime if and only if it divides v k. Make use of this to (re)confirm that M 5 =31and M 7 = 127 are primes, but that M 11 = 2047 is not.
Problem C1. (a) Find the area of the triangle whose sides have lengths 5, 6, 7. Leave your answer in the form of the square root of an integer. (b) Use Heron s method to find an approximation of the area of the triangle in (a) in the form of a fraction.
Problem C2. Find all shapes of Heron triangles that can be obtained by joining integer multiples of (3, 4, 5) and (5, 12, 13).
Problem C3. Find two indecomposable Heron triangles each having has its longest sides two consecutive integers, and the shortest side not more than 10.
Problem C4. The triangle (5, 29, 30; 72) is the smallest Heron triangle indecomposable into two Pythagorean triangles. Realize it as a lattice triangle, with one vertex at the origin.
Problem C5. The triangle (15, 34, 35; 252) is the smallest acute Heron triangle indecomposable into two Pythagorean triangles. Realize it as a lattice triangle, with one vertex at the origin.
Problem C6. For a given rational number k>2, verify that following triangle of rational sides has area k: 5k 2 4k +4, k 2 4 k(k 2 4k + 20), 2(k 2 4) k +2 2.
Problem D1. Use Archimedes method to find the area of the parabolic segment cut out by the parabola y 2 =4x and the line y =2x.
Problem D2. The constant π is the ratio of the circumference of a circle to its diameter. In the following questions give your answer first in an exact form involving square roots, then find its value using a calculator. (a) What is the value of π if the circle is taken as a regular octagon inscribed in it? (b) What is the value of π is the circle is taken as a regular octagon circumscribed in it? You may make use of tan 22 1 = 2 1 and cos22 1 2+ 2 =. 2 2 2
Problem D3. Let s n be the length of a side of a regular n-gon inscribed in a circle of radius 1. Beginning with s 6 =1, use the formula s 2n = ( ( s ) ) 2 ( n sn ) 2 2 + R R 2 2, 2 to calculate 48s 96 as an approximation of π, by using a calculator, each time, approximate all numbers to 8 digits after the decimal point. (If your calculator does not allow 8 digits after the decimal point, use 6 digits).
Problem D4. Here is Problem III.6 of Diophantus Arithmetica: To find three numbers such that their sum is a square and thesum of any pair is a square. Complete Diophantus solution to find an answer: Let the sum of all three be x 2 +2x +1, sum of first and second x 2, and therefore the third is. Let the sum of the second and the third be (x 1) 2. Therefore, the first = and the second =. But first + third = square, that is = square = 121, say. Therefore, x =, and the three numbers are, and.
Problem D5. Here is Problem III.8 of Diophantus Arithmetica: Given one number, to find three others such that the sum of any pair of them added to the given number gives a square, and also the sum of the three added to the given number gives a square. Follow Diophantus solution and complete it to find an answer: Given number 3. Suppose first required number + second = x 2 +4x +1, second + third = x 2 +6x +6, and sum of all three = x 2 +8x +13. Therefore, third =, second =, first. Also, first + third + 3 = a square, that is, =a square = 64, suppose. Hence, x =, and the three numbers are,,.
Problem D6. Here is Problem III.9 of Diophantus Arithmetica: Given one number, to find three others such that the sum of any pair of them minus the given number gives a square, and also the sum of the three minus the given number gives a square. Follow Diophantus solution and complete it to find an answer: Given number 3. Suppose first required number + second = x 2 +3, second + third = x 2 +2x +4, and sum of all three = x 2 +4x +7. Therefore, third =, second =, first. Lastly, first + third 3 =a square, that is, =a square = 64, say. Hence, x =, and the three numbers are,,.
Problem D7. Find all rational points on the ellipse 4x 2 +9y 2 =36.
Problem D8. Use the method of Arithmetica IV.24 to find a rational point on y 2 = x 3 +2, starting with ( 1, 1).
Problem D9. (Problem IX.7 of the Nine Chapters) When one end of a rope is tied to the top of a (vertical) pole, 3 chi rests on the ground. When stretched, the other end of (the rope) is 8 chi from the pole. How long is the rope? 2 2 Answer: 12 1 6 chi.
Problem D10. (Problem IX.13 of the Nine Chapters) A1zhang (= 10 chi) bamboo breaks and its top reaches ground, 3 chi from the bamboo. How tall is the broken bamboo? 3 3 Answer: 4 11 20 chi.
Problem D11. Use the method of the Nine Chapters to find, in each of the following, a right triangle with legs a, b, and hypotenuse c satisfying the conditions. (a) c + a = 40 and b = 21. [Give you answer as fractions, not in decimal.] (b) c + a =81and c + b =98.
Problem D12. (a) Imitate the solution of Fibonacci, Book of Squares, Problem 14, to find two adjacent chains of consecutive odd numbers with equal sums 2016. Hint: Use m =9, n =7. (b) Make use of this to construct a right triangle with rational sides and area 14.
Problem D13. (a) Draw a diagram to illustrate the relation (a + b + c) 2 =2(c + a)(c + b) for the sides of a right triangle. Here, c is the hypotenuse and a, b are the legs. (b) Make use of this relation to give an example of an integer right triangle with c + a is a square and c + b is twice of a square.
Problem E1. Decide if M 29 =2 29 1 = 536870911 is a prime. If not, give a complete factorization. Hint: Here are a few small primes of the form 58k +1: 59, 233, 349, 523, 929, 1103.
Problem E2. (a) Verify that (a 2 + b 2 )(x 2 + y 2 )=(ax by) 2 +(bx + ay) 2. (b) Make use of (a) to express 481 as a sum of two squares in two different ways. (c) Find all Pythagorean triangles (a, b, c) with a<band c = 481. Which of these are primitive?
Problem E3. For each of the following primes p 1(mod4), write p as a sum of two squares of integers by easy inspection. Make use of this to find a square root of 1 mod p. p x 2 + y 2 square root of 1 (mod p) 13 3 2 +2 2 73 97 113 137 149 157 173 181
Problem E4. In each of the following, q< p is a square root of 1 2 mod p. Use the Cornacchia algorithm to write p as a sum of two squares. (a) p = 1129, q = 168. (b) p = 1299709, q = 329008.
Problem E5. It is known that the continued fraction expansion of 14 is 14 = [3, 1, 2, 1, 6]. Make use of this to find (i) the fundamental solution of the Pell equation x 2 14y 2 =1; (ii) the complete solution of the Pell equation x 2 14y 2 =1in the form of a recurrence relation; (iii) the second and third smallest solution of the equation x 2 14y 2 =1.
Problem E6. It is known that the continued fraction expansion of 13 is 13 = [3, 1, 1, 1, 1, 6]. Make use of this to find (ii) the fundamental solution of the Pell equation x 2 13y 2 =1; (iii) the complete solution of the Pell equation x 2 13y 2 = 1 in the form of a recurrence relation; (iv) the second and third smallest solution of the equation x 2 13y 2 = 1.
Problem F1. Solve the cubic equation x 3 =6x +6.
Problem F2. Make use of the expansion of (x 2 +2x +3) 2 to solve the quartic equation x 4 +4x 3 + x 2 +36x 7=0.
Problem F3. Solve the following problem from Cardano s Ars Magna: There is a triangle the difference between the first and second sides of which is 1 and between the second the third side of which is also 1, and the area of which is 3. Find the sides of the triangle.
Problem F4. Solve the following problem from Cardano s Ars Magna: ABC is a right triangle and AD is perpendicular to its base. Its side, AB plus BD is 36, and AC plus CD is 24. Find its area.