Question

Show that if $p_k/q_k$ is a convergent of the simple continued fraction expansion of $\sqrt{d}$, then $|p^2_k − dq^2_k | < 1 + 2\sqrt{d}$.

score 213 · Answer 1 · 2016-11-11 00:13:17

213

Views

Show that if $p_k/q_k$ is a convergent of the simple continued fraction expansion of $\sqrt{d}$, then $|p^2_k − dq^2_k | < 1 + 2\sqrt{d}$.

Published on 11 Nov 2016 - 0:13

score 981 · Answer 2 · 2016-11-11 02:15:40

981

Views

Calculating denominators of convergents of a continued fraction

Published on 11 Nov 2016 - 2:15

score 63 · Answer 3 · 2016-11-11 05:43:53

63

Views

Periodic Continued Fraction Formula for $\sqrt{a^2+4}$ for any non-zero integer $a$

Published on 11 Nov 2016 - 5:43

score 112 · Answer 4 · 2016-11-11 13:06:48

112

Views

Existence of Irrational Number that has same $n$ digits of a given Rational Number.

Published on 11 Nov 2016 - 13:06

score 215 · Answer 5 · 2016-11-11 14:20:58

215

Views

Finding Irrational Approximation for a given Rational Number.

Published on 11 Nov 2016 - 14:20

score 63 · Answer 6 · 2016-11-12 21:34:48

63

Views

4-bonacci ratio Limit

Published on 12 Nov 2016 - 21:34

score 707 · Answer 7 · 2016-11-15 15:02:23

707

Views

On the uniqueness of a simple continued fraction

Published on 15 Nov 2016 - 15:02

score 27 · Answer 8 · 2016-11-16 12:53:23

27

Views

Why either of odd or even convergents lie on a line?

Published on 16 Nov 2016 - 12:53

score 158 · Answer 9 · 2016-11-16 14:14:03

158

Views

All solutions of $x^2 - dy^2 = -1$ are given by $x_n, y_n$ where $x_n + y_n \sqrt d = (x_1 + y_1 \sqrt d)^n$, with $n = 1,3,5,7,··$

Published on 16 Nov 2016 - 14:14

score 205 · Answer 10 · 2016-11-17 03:26:12

205

Views

Alternate proof of this identity for the generalized Harmonic number?

Published on 17 Nov 2016 - 3:26

score 385 · Answer 11 · 2026-03-27 04:38:34

385

Views

Can Every Irrational, but Non-Transcendental be Expressed as a Repeating Continued Fraction?

Published on 27 Mar 2026 - 4:38

score 79 · Answer 12 · 2016-11-22 06:58:00

79

Views

Proof about continued fractions

Published on 22 Nov 2016 - 6:58

score 50 · Answer 13 · 2016-11-27 16:32:06

50

Views

Show some forms of number can not have a purely periodic continued fraction expression

Published on 27 Nov 2016 - 16:32

score 99 · Answer 14 · 2016-11-27 18:44:15

99

Views

Continued fractions proofs

Published on 27 Nov 2016 - 18:44

score 182 · Answer 15 · 2026-03-26 06:00:53

182

Views

Rational Approximation Theorem

Published on 26 Mar 2026 - 6:00

List Question

Show that if $p_k/q_k$ is a convergent of the simple continued fraction expansion of $\sqrt{d}$, then $|p^2_k − dq^2_k | < 1 + 2\sqrt{d}$.

Calculating denominators of convergents of a continued fraction

Periodic Continued Fraction Formula for $\sqrt{a^2+4}$ for any non-zero integer $a$

Existence of Irrational Number that has same $n$ digits of a given Rational Number.

Finding Irrational Approximation for a given Rational Number.

4-bonacci ratio Limit

On the uniqueness of a simple continued fraction

Why either of odd or even convergents lie on a line?

All solutions of $x^2 - dy^2 = -1$ are given by $x_n, y_n$ where $x_n + y_n \sqrt d = (x_1 + y_1 \sqrt d)^n$, with $n = 1,3,5,7,··$

Alternate proof of this identity for the generalized Harmonic number?

Can Every Irrational, but Non-Transcendental be Expressed as a Repeating Continued Fraction?

Proof about continued fractions

Show some forms of number can not have a purely periodic continued fraction expression

Continued fractions proofs

Rational Approximation Theorem

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