The History of Mathematics: A Brief Course

182 7. ANCIENT NUMBER THEORY By brute-force enumeration of cases, the author concludes that there will be 377 pairs, and "in thi ...

QUESTIONS AND PROBLEMS 183 7.4. Draw dot figures for the first five heptagonal and octagonal numbers. What kind of figure would ...

184 7. ANCIENT NUMBER THEORY 7.10. Take as a unit of time Ô = ^ of a year, about 37 hours, 18 minutes, say a day and a half in c ...

QUESTIONS AND PROBLEMS 185 The first few partial fractions here give 1 Ú 1 + 2+T = 3, 1 2 8 2 + —Ã =^23 = 3 =^2 -^666 - 2 1 _ ~ ...

186 7. ANCIENT NUMBER THEORY y = 2uv, and æ = u^2 + v^2. [Hint: Start from the fact that x^2 = (z — y)(z + y), so that æ - y = a ...

Chapter 8. Numbers and Number Theory in Modern Mathematics Beginning with the work of Fermat in the seventeenth century, number ...

188 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS See the website http://www.euler2007.com/. was given by Euler in 1738. Ac ...

MODERN NUMBER THEORY 189 with an odd number of sides will be those whose number of sides is a product of distinct numbers from ...

190 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS Euler conjectured what we now know as the law of quadratic reciprocity: G ...

MODERN NUMBER THEORY 191 construct all solutions of the Pell equation Dx^2 ± 1 = y^2 (see Scharlau and Opolka, 1985, pp. 45-56 ...

192 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS Street in Gottingen named after Gauss. The 1801 work Disquisitiones arith ...

MODERN NUMBER THEORY 193 until Dirichlet restored its simplicity by going back very nearly to Legendre's original construction ...

194 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS subjects had been drawing closer together ever since Euler began his stud ...

MODERN NUMBER THEORY 195 other zeros have real part equal to 5. This Riemann hypothesis forms one of the still-outstanding uns ...

196 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS that complex numbers of the form an + á\è + • • • + áç-éèç~é, where èç = ...

NUMBER SYSTEMS 197 Chebyshev in his 1859 paper on ð(χ), even though he was in close contact with Dirichlet, and Chebyshev's pa ...

198 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS China. Chinese counting rods were red or black according as the numbers r ...

NUMBER SYSTEMS 199 operations, for example, considering the appropriate value of a fraction such as (5x^2 + Ax)/(3x^2 — 2x) wh ...

200 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS equilateral and oblong. The former class consists of the squares of ratio ...

NUMBER SYSTEMS 201 Side é—é—é—é—é Diagonal é—é—é—é—é—é—i— Side é é é é é é Diagonal é—é—é—é—\—é—é—é—é FIGURE 1. Diagonal and s ...