Proof without words: Periodic continued fractions

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Abstract

In this paper, We give a generalization the resut of Roger B. Nelsen, by giving a closed form expression for x =$[a0,a1,⋯,ak,b1,⋯bm¯],$

Abstract

In this paper, We give a generalization the resut of Roger B. Nelsen, by giving a closed form expression for x =$[a0,a1,⋯,ak,b1,⋯bm¯],$

1 Introduction

Let x := x0 be a real number. Set a0 = [x], the greatest integer in x and $1x0−a0$its complete quotients.

Set ai = [xi], the greatest integer in xi and $xi+1=1xi−afor all i≥1.$Then,

$x=a0+1a1+1a2+1…$

The algorithm stops after finitely many steps if and only if x is rational. The above expansion is called The simple continued fraction of x. It is customarily written x = [a0,a1, . . . ,an,. ].

We call convergents of x the reduced fractions difined by:

$p0q0=a0,$
$p1q1=a0+1a1,… ,pnqn=a0+1a1+1a2+1a3+⋯1an,⋯.$

If there exists k ≥ 0 and m > 0 such that whenever r > k, we have ar = ar+m, the continued fraction is said periodic, with period (b1, . . . ,bm) = (ak+1, . . . ,ak+m) and pre-period (a0,a1, . . . ,ak), which can be written for simplicity $x=[a0,a1,⋯ak,b1 ,⋯bm¯].$These so-called periodic continued fractions are precisely those that represent quadratic irrationalities.

We find a closed form expression for $x=[a0,a1,⋯ak,b1 ,⋯bm¯]$, which generalized a previous resut of Roger B. Nelsen.

2 Main result

Lemma 1

Let x > 0 such that $x=a+bcx,then x =12(a2+a2+4bc)$

Proof. Consider the Following figure:

We have

Lemma 2

$Ifx=a0,a1,⋯,an¯,thenx=pn−qn−1qn+pn−1qnx.$

Proof. We have $x=a0,a1,⋯,an¯=a0,a1,⋯an,x=pnx−pn−1qnx−qn−1.Then,qnx2=pn−qn−1+pn−1.which$

gives $x=pn−qn−1qn+pn−1qnx.$Completing the proof.

Theorem 3

The periodic continued fraction $[a0,a1,⋯,an¯]$equals

$12pn−qn−1qn2+pn−qn−1qn2+4pn−1qn.$

Corollary 4

(Theorem  ). . The periodic continued fraction $[a,b¯]$equals

$12(a2+a2+4ab).$

Corollary 5

The periodic continued fraction $[a,b,c¯]$equals

$12a+c-bbc+12+a+c-bbc+12+4ab+1bc+1.$

Example 6

As examples, notice that $[1¯]=[1,1,1¯]=12(1+5),[a¯]=[a,a¯]=[a,a,a¯]=12(a2+a2+4),$$[3,1,2¯]=12(1009+1489).$

Corollary 7

Let $x=[a0,a1,…,ak,b1,⋯bm¯],$be a periodic continued fraction, with period (b1, . . . ,bm) and pre-period (a0,a1, . . . ,ak).

Note $piqi=[a0,a1,⋯,ai],for all 0≤i≤k and p′jq′j=[b1,⋯bj]for all 0 ≤j≤m.$Then,

$x=pk(12[(p′m−q′m−1q′m)2+(p′m−q′m−1qn)2+4p′m−1q′m])+pk−1qk(12[(p′m−q′m−1q′m)2+(p′m−q′m−1qn)2+4p′m−1q′m])+qk−1.$

Example 8

As examples, notice that

$[1,2,3,4,5,2,1,1,1,4¯]=2257+431577+30,[1,2,2,n,1,2n¯]=7n2+2n+35n2+2n+2,[1,2,2,1,4,n,n,2n¯]=57n2+2+1033n2+2+7.$

3 Conclusions

We find a closed form expression $forx=a0,a1,⋯,ak,b1,⋯,bm¯,$which generalized a previous resut of Roger B. Nelsen.

Acknowledgment

The author would like to thank the editor and the anonymous referee who kindly reviewed the earlier version of this manuscript and provided valuable suggestions and comments.

Communicated Juan Luis García Guirao

References



Roger B. Nelsen (2018) Periodic Continued Fractions Via a Proof Without Words, Mathematics Magazine, 91:5, 364-365,



Roger B. Nelsen (2018) Periodic Continued Fractions Via a Proof Without Words, Mathematics Magazine, 91:5, 364-365,

Applied Mathematics and Nonlinear Sciences   