On the continued fraction expansions of (1

Comptes Rendus

Mathématique

Stéphane R. Louboutin

On the continued fraction expansions of (1

)/2 and

Volume 359, issue 9 (2021), p. 1201-1205 Published online: 25 November 2021 https://doi.org/10.5802/crmath.266

This article is licensed under the

Creative Commons Attribution4.0International License.

http://creativecommons.org/licenses/by/4.0/

LesComptes Rendus. Mathématiquesont membres du Centre Mersenne pour l’édition scientifique ouverte

www.centre-mersenne.org e-ISSN : 1778-3569

2021, 359, n9, p. 1201-1205

https://doi.org/10.5802/crmath.266

Number theory /Théorie des nombres

On the continued fraction expansions of (1 + p

p q )/2 and p p q

Stéphane R. Louboutin^a

aAix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France E-mail:[email protected]

Abstract.The evenness and the values modulo 4 of the lengths of the periods of the continued fraction expansions ofpp andp

2pforp≡3 (mod 4) a prime are known. Here we prove similar results for the continued fraction expansion ofpp q, wherep,q≡3 (mod 4) are distinct primes.

Mathematical subject classification (2010).11A55, 11R11.

Manuscript received 21st June 2021, accepted 27th August 2021.

1. Introduction

Letαbe a real quadratic irrational number. Its continued fraction expansionα=[a₀,a₁,a₂, . . . ] is periodic, i.e. there existsk ≥0 andl ≥1 such thata_i+l =a_i fori ≥k. In that case we write α=[a₀, . . . ,a_k−1,a_k, . . . ,a_k+l−1]. The least suchlis called the length of the period of the periodic continued fraction expansion ofα. The evenness of the length of the period of the continued fraction expansion ofppforp≡3 (mod 4) a prime is well known. In [8] we determined its value modulo 4 and gave a similar result forp

2p:

Theorem 1. Take d=p or d=2p, where p≡3 (mod 4)is a prime integer. Let l≥1be the length of the period of the periodic continued fraction expansionp

d=[a₀,a₁, . . . ,a_l]. Then, (i) a₀= bp

dc, a_l=2a₀and a_k=a_l−kfor1≤k≤l−1, (ii) l=2L is even and L is even if and only if p≡7 (mod 8), (iii) a_l/2=a_Lis the integer in{a₀−1,a₀}of the same parity as d .

This behavior in the case ofd=phad already been proved in [3, Corollary 2 p. 2071]. Our proof was different and applied both tod=pandd=2p. It was based on the arithmetic of quadratic number fields and their ideal class groups in the narrow sense (as in [6] and [7]). LetI be an integral ideal of the ring of algebraic integersZK of a real quadratic number fieldK. Recall that Iis principal if and only if there existsα∈ZKsuch thatI=αZK, whereasI is principal in the narrow sense if there exists a totally positive elementα∈ZK such thatI=αZK. Here, bearing on a similar approach, we prove:

1202 Stéphane R. Louboutin

Theorem 2. Let p,q be two prime integers equal to3 (mod 4), with3≤p<q. Let l≥1be the length of the period of the periodic continued fraction expansion(1+pp q)/2=[a₀,a₁, . . . ,a_l].

Then,

(i) a_l=2a₀−1and a_k=a_l−kfor1≤k≤l−1, (ii) l=2L is even and(−1)^L=³_p

q

´

(Legendre’s symbol), (iii) a_l/2=a_Lis the unique odd integer in{bp

q/pc −1,bp q/pc}.

Theorem 3. Let p,q be two prime integers equal to3 (mod 4), with3≤p<q. Let l≥1be the length of the period of the periodic continued fraction expansionpp q=[a₀,a₁, . . . ,a_l]. Then,

(i) a₀= bppqc, a_l=2a₀and a_k=a_l−kfor1≤k≤l−1, (ii) l=2L is even and(−1)^L=

³_p

q

´

(Legendre’s symbol), (iii) a_l/2=a_L=2bp

q/pcis even.

Part of Theorem 3 was proved in [10, Corollary 1], [2, Theorem 2] and [1], but notice that point (iii) of Theorem 3 is much more precise than [1, Theorem 1.2].

2. On the continued fraction expansions of some real quadratic irrational numbers

d)/Q for some non-square integerd>1, someP∈Zand someQ∈Z\ {0} dividingd−P². Thenωis calledreducedifω>1 and−1/ω⁰>1, whereω⁰=(P−p

d)/Qis the conjugate ofωinQ(p

d). Hence,ωis reduced if and only ifP+p

d>Q>p

d−P>0, which implies 0<Q<2p

d,|P| <p d, 2p

d/Q−1<ω<2p d/Q andbωc ∈{b2p

d/Qc −1,b2p d/Qc}.

(ii). Thecontinued fraction expansionω0=[a0,a1, . . . ] ofω0=(P0+p

d)/Q0withP0,Q0∈Z, and Q06=0 dividingd−P₀², can be computed inductively by writingωk=[a_k, . . . ] asωk=(P_k+p

d)/Q_k, where theP_k,Q_k∈ZwithQ_k6=0 dividingd−P_k²are inductively computed, usinga_k= bωkcand ωk=a_k+1/ωk+1, byP_k+1=a_kQ_k−P_kandQ_k+1=(d−P_k+1² )/Q_k=(d−P_k²)/Q_k+2a_kP_k−a_k²Q_k. (HenceQ₁is a non-zero rational integer,Q_k+1=Q_k−1+2a_kP_k−a_k²Q_kfork≥1 and theQ_k’s are non-zero rational integers, by induction onk.)

(iii). Assume thatω0=(P0+p

d)/Q0is reduced. Usingωk=a_k+1/ωk+1, we obtain that all the ωk’s are reduced, by induction. Hence 0<Q_k <2p

d and|P_k| <p

d fork ≥0 and there are only finitely many pairwise distinctωk’s. It follows thatωm =ωn for somem>n ≥0, which impliesωk+l =ωkanda_k+l=a_kfork≥b, wherel:=m−n≥1. Hence, the continued fraction expansion ofω0isl-periodic. In fact is purely periodic, which we writeω0=[a₀, . . . ,a_l−1], i.e.

ωk+l=ωkanda_k+l=a_kfork≥0, wherel:=m−n≥1. (Notice thatωk+l=ωkandk≥1 imply ωk−1−a_k−1=1/ωk=1/ωk+l=ωk+l−1−a_k+l−1, hence implyωk+l−1−ωk−1=a_k+l−1−a_k−1∈Z andωk+l−1−ωk−1=ω⁰_k+l−1−ω⁰_k−1∈(−1, 1)∩Z, hence implyωk+l−1=ωk−1.) The least suchl≥1 is calledthe lengthof the purely periodic continued fraction expansion of the reduced quadratic irrational numberω0.

In that case−1/ω⁰₀=[a_l−1, . . . ,a₀] (e.g. see [4, XV page 311]).

(iv). If ω0 =[a0,a1, . . . ,a_l−1]∈ Q(p

d) is reduced, using ωk = a_k+1/ωk+1 we obtain Mk := Z+Zωk=Z+Zω⁻¹_k+1=ω⁻¹_k+1Mk+1andM0=ω⁻¹₁ M1=ω⁻¹₁ ω⁻¹₂ M2= · · · =ε⁻¹Ml =ε⁻¹M0, where ε=ω1ω2. . .ωl=ω0ω1. . .ωl−1. Therefore,εis a unit of normN(ε)=Q_l−1

k=0(ωkω⁰_k)=(−1)^l of the Z-moduleM0=Z+Zω0⊆Q(p

d) (asωk>1 and−1/ω⁰_k>1).

(v). See [4, p. 305–322], [5, Chapter 10] and [9] for more information on continued fractions.

C. R. Mathématique—2021, 359, n9, 1201-1205

3. Proof of Theorem 2

Letd≡1 (mod 4) be a non-square integer, withd≥5. Letg⁰≥1 be the unique odd integer in [p

d−2,p

d). Thenω0=(P₀+p

d)/Q₀=(g⁰+p

d)/2 is reduced. Its continued fraction expansion ω0=[g⁰,a1, . . . ,a_l−1] is purely periodic andω1=[a1, . . . ,a_l−1,g⁰]=1/(ω0−g⁰)=2/(p

d−g⁰)=

−1/ω⁰₀=[a_l−1, . . . ,a₁,g⁰]. Hence,a_k=a_l−kfor 1≤k≤l−1. UsingQ₀=2, the oddness ofP₀=g⁰, the evenness ofQ1=(d−P²₀)//Q0=(d−g⁰²)/2 and the identitiesQ_k+1=Q_k−1+2a_kP_k−a_k²Q_kfor k≥1 andP_k+1=a_kQ_k−P_kfork≥0, we obtain that theQ_k’s are even and theP_k’s are odd for k≥0. Consequently, ifdis square-free thenM0is equal to the ring of algebraic integersZ_Kof the real quadratic number fieldK=Q(p

d) and theZ-modules Ik:=(Q_k/2)Mk=(Q_k/2)Z+P_k+p

d

2 Z=(Q_k/2)ω1. . .ωkM0=αkZK

are primitive, principal, integral ideals of normsQ_k/2 of the real quadratic number fieldQ(p d), whereαk=(Q_k/2)ω1. . .ωk∈Ik ⊆Z_K is an algebraic integer of norm (−1)^k(Q_k/2) (recall that ωk>1 and−1/ω⁰_k>1 fork≥0). Hence,Ikis principal in the narrow sense if and only ifkis even.

Now, assume that d is divisible by a prime p ≡ 3 (mod 4). Since the congruence x²−d y² ≡ −4 (modp) has no solution in rational integers, any algebraic unit of Q(p

d) has norm +1. The algebraic unit ε = ω0ω1. . .ωl−1 := Z+Zω0 being of norm (−1)^l, l =2L is even, ω0 = [g⁰,a1, . . . ,aL−1,aL,aL−1, . . . ,a1], ωL = [aL, . . . ,a1,g⁰,a1, . . . ,aL−1] and ωL+1 = [a_L−1, . . . ,a₁,g⁰,a₁, . . . ,a_L] = −1/ω⁰_L. Hence, −1 = ωL+1ω⁰_L = ^PL+1+

pd QL+1

PL−p d

QL , which implies PL+1=PL. SincePL+1=aLQL−PL, we havePL+1=aL(QL/2) andaLis odd. Moreover,

d−P_L²₊₁=d−a²_L(Q_L/2)²=4(Q_L/2)(Q_L+1/2).

Hence,Q_L/2 dividesd. Finally,ωL=(P_L+p

d)/Q_L being reduced, we have 1<Q_L<2p d and a_L= bωLc ∈{b2p

d/Q_Lc,b2p

d/Q_Lc −1}, and we obtain the following Proposition and Corollary from which Theorem 2 follows:

Proposition 4. Let d ≡1 (mod 4) be a square-free integer, with d ≥5 such that at least one prime p≡3 (mod 4)divides d . Let g⁰≥1be the unique odd integer in the interval[p

d−2,p d).

Setω0=(g⁰+p

d)/2. l ≥1be the length of the period of the purely periodic continued fraction expansionω0=[g⁰,a₁, . . . ,a_l−1]. Then

(i) a_k=a_l−kfor1≤k≤l−1;

(ii) l=2L is even;

(iii) Q_L/2divides d and1<Q_L/2<p d ; (iv) a_Lis odd and a_L= bωLc ∈{b2p

d/Q_Lc,b2p

d/Q_Lc −1};

(v) The integral idealIL=(QL/2)Z+^PL+₂^pdZof norm QL/2is principal and L is even if and onlyILis principal in the narrow sense.

Corollary 5. Let p,q be two prime integers equal to3 (mod 4), with3≤p<q. Take d=p q≡1 (mod 4). Then Q_L/2=p. Hence,ILis the prime ramified idealP of norm p of the ring of algebraic integers of the real quadratic fieldQ(p

d)and aLis the unique odd integer in{bp

q/pc,bp

q/pc−1}.

Moreover,P is principal in the narrow sense if and only if³_p

q

´

= +1.

Proof. LetP andQbe the prime ideals above p andq, respectively. HenceP =I =(α) is principal andP Q=(p

d) is also clearly principal. Sincep

d∈P Q⊆P =(α), we havep d=αβ for some algebraic integerβ. HenceQ=(β) is also principal. SinceN(α)N(β)=N(αβ)=N(p

d)=

−d<0, only one of the two principal idealsP orQis principal in the narrow sense. IfP =(α) is principal in the narrow sense, withα=(x+yp

d)/2 such thatp=N(α)=(x²−p q y²)/4, then pdividesx=p X, 4=p X²−q y²and³_p

q

´

= +1. IfP is not principal in the narrow sense, then Q=(β) is principal in the narrow sense, withβ=(x+yp

d)/2 such thatq=N(β)=(x²−p q y²)/4.

Hence,qdividesx=q X, 4=q X²−p y²and³_−p

q

´

= −

³_p

q

´

= +1.

1204 Stéphane R. Louboutin

4. Proof of Theorem 3

Letd≡1 (mod 4) be a non-square integer, withd≥5. Setg= bp

dc. Thenω0=(P₀+p

d)/Q₀= g+p

dis reduced. SinceM0=Z[ω0]=Z[p

d] is not the ring of algebraic integers ofQ(p d), the proof of Theorem 3 is a little more tricky than the one of Theorem 2. Here again, the continued fraction expansionω0=[2g,a1, . . . ,a_l−1] is purely periodic andω1=[a1, . . . ,a_l−1, 2g]=1/(ω0− 2g)=1/(p

d−g)= −1/ω⁰₀=[a_l−1, . . . ,a1, 2g]. Hence,a_k =a_l−k for 1≤k ≤l−1. Suppose that we hadQ_n ≡2 (mod 4) for somen≥0. ThenP_n+1would be odd andQ_n+1would be even, as Q_nQ_n+1=d−P_n²₊₁. Therefore, all theQ_k’s would be even fork≥n, asQ_k+1=Q_k−1+2a_kP_k−a²_kQ_k fork≥1, hence fork≥0, by pure periodicity of the continued fraction expansion ofω0. SinceQ₀ is odd, we deduce thatQ_k6≡2 (mod 4) fork≥0.

Now, assume thatdis divisible by a primep≡3 (mod 4). As above,l=2Lis even and 2P_L+1=a_LQ_L, and 4d−a²_LQ²_L=4Q_LQ_L+1.

Hence,Q_L divides 4d and 4 does not divideQ_L and we obtain the following Proposition from which Theorem 3 follows, by Corollary 5:

Proposition 6. Let d≡1 (mod 4)be a square-free integer, with d≥5such that at least one prime p≡3 (mod 4)divides d .

Setω0=g+p

d , where g= bp

dc. Let l ≥1be the length of the period of the purely periodic continued fraction expansionω0=[2g,a₁, . . . ,a_l−1]. Then

(i) a_k=a_l−kfor1≤k≤l−1;

(ii) l=2L is even;

(iii) Q_Ldivides2d and1<Q_L<2p d ; (iv) aL= bωLc ∈{b2p

d/QLc,b2p

d/QLc −1};

(v) if d =p q, where p,q are prime numbers equal to3 modulo4with p <q, then a_L = 2bp

q/pc, Q_L =p, the prime idealP of norm p of the ring of algebraic integers of the real quadratic fieldQ(p

d)is principal and L is even if and only ifP is principal in the narrow sense.

Proof. It remains to prove point (v). SinceQLdivides 2p q,QL6≡2 (mod 4) andQL<2pp q, we haveQ_L∈{p,q}. SinceQ_L=qwould yield the contradiction 4qQ_L+1=4d−a²_LQ²_L≤4p q−4q²<0, we haveQL=p. Hence,aLis even, as 2PL=aLQL, andaL∈{b2xc,b2xc −1}, wherex=p

d/QL. Sinceb2xc ∈{2bxc, 2bxc +1} forxreal, we have thata_L is even anda_L∈{2bxc −1, 2bxc, 2bxc +1}.

Therefore,a_L=2bxc =2bp

d/Q_Lc =2bp q/pc. Finally, setβL=Q_Lω1. . .ωL. Then

JL:=βLZ[p

d]=βLM0=QLω1. . .ωLM0=QLML=QLZ+(PL+p

d)Z⊆Z[p d].

Hence,βL=x+yp d∈Z[p

d] and {QL,PL+p

d} and {βL,βL

pd}={x+yp

d,d y+xp

d} are two Z-bases ofJLand the change of basis matrix

A= Ã_x−yP

L Q_L

d y−xP_L Q_L

y x

!

is in M₂(Z) and of determinant ±1, i.e. ±1 =(x²−d y²)/Q_L =N(βL)/p. Therefore, N(βL)= (−1)^Lp, with βL ∈Z[p

d]. It follows that the prime ideal P of the ring of algebraic integers ZK=Z[(1+p

d)/2] ofK=Q(p

d) lying abovep is principal and equal to (βL) and thatP it is

principal in the narrow sense if and only ifLis even.

C. R. Mathématique—2021, 359, n9, 1201-1205

References

[1] S. Das, D. Chakraborty, A. Saikia, “On the period of the continued fraction ofppq”,Acta Arith.196(2020), no. 3, p. 291-302.

[2] C. Friesen, “Legendre symbols and continued fractions”,Acta Arith.59(1991), no. 4, p. 365-379.

[3] E. P. Golubeva, “Quadratic irrationals with fixed period length in the continued fraction expansion”,J. Math. Sci., New York70(1994), no. 6, p. 2059-2076.

[4] H. Hasse,Vorlesungen über Zahlentheorie, Grundlehren der Mathematischen Wissenschaften, vol. 59, Springer, 1964.

[5] L. K. Hua,Introduction to number theory, Springer, 1982, translated from the Chinese by Peter Shiu.

[6] S. R. Louboutin, “Continued fractions and real quadratic fields”,J. Number Theory30(1988), no. 2, p. 167-176.

[7] ——— , “Groupes des classes d’idéaux triviaux”,Acta Arith.54(1989), no. 1, p. 61-74.

[8] ——— , “On the continued fraction expansions ofppandp

2pfor primesp≡3 (mod 4)”, inClass groups of Number fields and related topics, Springer, 2020, p. 175-178.

[9] O. Perron,Die Lehre von den Kettenbrüchen. Band I. 3. erweiterte und verbesserte Aufl., Teubner, 1954.

[10] A. J. van der Poorten, P. G. Walsh, “A note on Jacobi symbols and continued fractions”,Am. Math. Mon.106(1999), no. 1, p. 52-56.