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 1x0a0its complete quotients.

Set ai = [xi], the greatest integer in xi and xi+1=1xiaforalli1.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,thenx=12(a2+a2+4bc)

Proof. Consider the Following figure:

We have 2xa2=a2+4bc,then x=12a2+a2+4bc.

Lemma 2

Ifx=a0,a1,,an¯,thenx=pnqn1qn+pn1qnx.

Proof. We have x=a0,a1,,an¯=a0,a1,an,x=pnxpn1qnxqn1.Then,qnx2=pnqn1+pn1.which

gives x=pnqn1qn+pn1qnx.Completing the proof.

Theorem 3

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

12pnqn1qn2+pnqn1qn2+4pn1qn.

Corollary 4

(Theorem [1] ). . 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],forall0ikandpjqj=[b1,bj]forall0jm.Then,

x=pk(12[(pmqm1qm)2+(pmqm1qn)2+4pm1qm])+pk1qk(12[(pmqm1qm)2+(pmqm1qn)2+4pm1qm])+qk1.

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

[1]

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

[1]

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

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