Number Theory: An Introduction to Mathematics
186 IV Continued Fractions and Their Uses Thus we now assumeμ=0. Sinceq≤qn,λandμcannot both be positive and hence, sinceq>0, ...
2 Diophantine Approximation 187 The complete continued fraction expansion ofπis not known. However, it was discovered by Cotes ( ...
188 IV Continued Fractions and Their Uses As an application of Proposition 4 we prove Proposition 5Let d be a positive integer w ...
2 Diophantine Approximation 189 and hence qm^2 − 2 sqm− 1 qm+qm^2 − 1 ≤ 0. Furthermore, this inequality also holds whenqm− 1 ,qm ...
190 IV Continued Fractions and Their Uses thenan=1 for all largen. The constant √ 8 is again best possible, since a similar argu ...
3 Periodic Continued Fractions 191 To establish this, putqθ=p+δ,wherep∈Zand|δ|≤ 1 /2. Then |e^2 πiqθ− 1 |= 2 |sinπqθ|= 2 |sinπδ| ...
192 IV Continued Fractions and Their Uses Ifζis a quadratic irrational, we define theconjugateζ′ofζto be the other root of the q ...
3 Periodic Continued Fractions 193 Proof Suppose first thatξ=[a 0 ,...,ah− 1 ] has a periodic continued fraction expan- sion. Th ...
194 IV Continued Fractions and Their Uses The proof of Proposition 7 shows that the period is at mostg(g+ 1 )/2 and thus is cert ...
4 Quadratic Diophantine Equations 195 But − 1 /ξ 1 ′=ξ+a 0 =[2a 0 ,a 1 ,...,ah]. Comparing this with the previous expression, we ...
196 IV Continued Fractions and Their Uses where x=x 1 x 2 +dy 1 y 2 , y=x 1 y 2 +y 1 x 2. (In fact, Brahmagupta’s identity is ju ...
4 Quadratic Diophantine Equations 197 Proposition 10Let d be a positive integer which is not a perfect square. Suppose ξ= √ d ha ...
198 IV Continued Fractions and Their Uses The least solution in positive integers, obtained by takingk=1, is called thefunda- me ...
4 Quadratic Diophantine Equations 199 of (10). Since u=x 0 u 0 −dy 0 v 0 =x 0 u 0 −[(x 02 − 1 )(u^20 −m)]^1 /^2 > 0 , we must ...
200 IV Continued Fractions and Their Uses and then from Proposition 7 thatξm+ 1 =− 1 /ξm′+ 1. But, by the proof of Proposi- tion ...
5 The Modular Group 201 Since the second term on the right is divisible byp^2 ,wemusthavexv =εyuor xu=−εyv. Evidentlyε=1inthefir ...
202 IV Continued Fractions and Their Uses Furthermore, if g(z)=(a′z+b′)/(c′z+d′) is another modular transformation, then the com ...
5 The Modular Group 203 SinceS−^1 =Sand(Tn)−^1 =T−n, it follows that g=Tm^1 S···TmkSTmorg=Tm^1 S···TmkTm. The proof of Proposi ...
204 IV Continued Fractions and Their Uses Proof For anyz∈Cwe writez=x+iy,wherex,y∈R. We show first that no two points ofFare equ ...
5 The Modular Group 205 This is illustrated in Figure 2, where the domaing(F)is represented simply by the group elementg. There ...
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