Aneesa Mughal Paper 07-11-2012

Transitive G-subsets of an invariantsubset Q∗(

√p ) of Q(

√p ), p ≡ 1(mod 4),

under the Modular Group Action

M. Aslam Malik ∗and Aneesa Mughal †

Department of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan.

Abstract

In this paper we investigate the action of the modular group PSL(2,Z)on the real projective line with an emphasis on finding orbit structureof a G-set

Q∗(√

n ) = {a +√

n

c: a, c 6= 0, b =

a2 − n

c∈ Z and (a, b, c) = 1}

where n = k2m, k ∈ N . Q(√

m)\Q is the disjoint union of Q∗(√

k2m )for all k ∈ N.We classify the G-orbits of Q∗(

√p ), p ≡ 1(mod 4), and find their

ambiguous lengths as a function of p. Specifically it is proved that thenumber oG(p) of all G-orbits of Q∗(

√p ) is congruent to 0(mod 2).

AMS Mathematics Subject Classification (2000): 05C25, 11E04, 20G15Keywords: Real quadratic irrational number; Modular Group; Linear-

fractional transformations; Transitive G-subset.

∗[email protected]†[email protected]

1

1 Introduction

It is well known that Q(√

m) = {u + v√

m : u, v ∈ Q} is a real quadraticfield for square free m > 0. An element α of Q(

√m) \ Q is called a real

quadratic irrational number and G = 〈x, y : x2 = y3 = 1〉 represents themodular group for x(z) = −1

z, y(z) = z−1

zas Mobius transformations. Q.

Mushtaq in 1988 has shown that every real quadratic irrational number canbe uniquely expressed as a+

√n

c, where n = k2m, k ∈ N , and (a, a2−n

c, c) = 1.

He also proved that the set

Q∗(√

n ) = {a +√

n

c: a, c 6= 0, b =

a2 − n

c∈ Z and (a, b, c) = 1}

is a proper G-subset of Q(√

m ) for all k ∈ N . Higman et al. have provedin (1988) that the action of the modular group G on Q ∪ {∞} is transitive.Since the set of all real quadratic irrational numbers in Q(

√m ) is the disjoint

union of all Q∗(√

k2m ), that is, Q(√

m) \Q = ∪k∈NQ∗(√

k2m ).The action of G on Q∗(

√n ) was discussed by M. Aslam Malik et al.

(2000, 2003), Q. Mushtaq (1988,1998) and S. Anis and Q. Mushtaq (2008)and it was proved in (M. Aslam Malik et al., 2005) that G acts intransitively

on Q∗(√

n ), n 6= 2. For α = a+√

nc

∈ Q∗(√

n ), if α and its algebraic

conjugate α = −a+√

n−c

, as real numbers, have different signs, then α is calledan ambiguous number. These ambiguous numbers play a significant role todetermine the structure of G-orbits of Q∗(

√n ). The coset diagrams are

used to investigate local-global relationship between real quadratic irrationalnumbers and the elements of G. By using coset diagrams, Q. Mushtaq, in1988, has shown that for each non-square n the set

Q∗1(√

n ) = {a +√

n

c∈ Q∗(

√n ) : a2 < n}

is finite and that part of the coset diagram consisting of the elements ofQ∗

1(√

n ) forms a single circuit (closed path) and it is the only circuit con-tained in the coset diagram for the orbit αG, α ∈ Q∗(

√n ). Thus the number

oG(n) of all G-orbits of Q∗(√

n ) is equal to the number of circuits in thecoset diagram under the action of G on Q∗(

√n ) (M. Aslam Malik et al.

1995).

A circuit is a closed path of edges and triangles in the coset diagram for theG-orbit αG, α ∈ Q∗(

√n ). If n1, n2, . . . , nk is a sequence of positive integers

2

then by a circuit of the type (n1, n2, . . . , n2k), we shall mean the circuit inwhich n1 triangles have one vertex inside (outside) the circuit and n2 triangleshave one vertex outside (inside) the circuit and so on n2k triangles have onevertex outside (inside) the circuit. This circuit induces an element

g = (yx)n2k . . . (yx)n3(y−1x)n2(yx)n1 (1)

of G and fixes a particular vertex of a triangle lying on the circuit (Q. Mush-taq, 1998). The set of ambiguous numbers in the orbit αG, α ∈ Q∗(

√n ) is

denoted by (αG)amb and the ambiguous length of αG is denoted by |αG|amb.It is clear from (1) that, |αG|amb = 2(n1 +n2 + . . .+n2k). Thus, it motivatesto know the cardinality of Q∗

1(√

n ) on one hand and the number of circuitsformed by these numbers on the other hand.The exact number |Q∗

1(√

n )| has been determined in (M. Aslam Malik et al.1995 and S. M. Husnine et al. 2005) as a function of n.

The G-subsets, not necessarily transitive, of Q∗(√

n ) have been exploredin (M. Aslam Malik et al. 2005, M. Aslam Malik and M. Asim Zafar 2011)by using the notion of congruences and quadratic residues. Thus it becomesinteresting to explore the transitive G-subsets (G-orbits) of Q∗(

√n ) and to

determine the number oG(n) of all G-orbits of Q∗(√

n ). It is also interestingto find the formula for calculating the ambiguous length |αG|amb of the G-orbit αG, where α ∈ Q∗(

√n ), as a function of n.

Throughout this paper G stands for the modular group, p for prime, n fornon-square positive integer and α = a+

√n

c∈ Q∗(

√n ). In this paper we study

the structure of the circuits formed by the elements of Q∗1(√

n ). In Section 2we have determined the formulae to calculate the ambiguous lengths of theG-orbits of Q∗(

√p ) in terms of p. These ambiguous lengths of the G-orbits

help us to find the remaining orbits of Q∗(√

p ), p ≡ 1(mod 4) and classifythem as well. In Section 3 we concentrate on the distribution of the ambigu-ous elements of Q∗(

√p ) in the G-orbits and have been able to prove that if

p ≡ 1(mod 4), then (α)G∩(α)G = ∅ for all α ∈ Q∗(√

p )\((√

p )G∪(1+√

p

2)G).

Specifically, we have proved that oG(p) ≡ 0(mod 2) for p ≡ 1(mod 4).Throughout the paper we have employed geometric insight to reduce thecomputations.

The following results of (M. Aslam Malik et al. 1995, 2000 and 2005) willbe used in the sequel.

3

Lemma 1.1 (M. Aslam Malik et al. 1995) Let m be a square-free posi-tive integer. Then

|Q∗1(√

m )| = τ ∗(m) = 2τ(m) + 4

b√mc∑a=1

τ(m− a2).

Lemma 1.2 (M. Aslam Malik et al. 2000) Let p ≡ 1(mod 4) such that p =

a2+c2. Then there are exactly eight ambiguous numbersa+√

p

±c,−a+

√p

±c,

c+√

p

±a,−c+

√p

±a

of Q∗(√

p ) which are mapped onto their conjugates under x.Lemma 1.3 (M. Aslam Malik et al. 2000) Let α ∈ Q∗(

√n). Then αG = (α)G

if and only if there exists an element β in αG such that x(β) = β.Lemma 1.4 (M. Aslam Malik et al. 2000) Let k ∈ N and α ∈ Q∗(

√n )

Then:1. (yx)k(α) = k + α. 2. (xy2)k(α) = −k + α.3. g(α) = g(α) for all g ∈ G. 4. |αG|amb = |αG|amb.5. x(−α) = −x(α) 6. y(−α) = 2− y(α)7. xy2(−α) = −[yx(α)] 8. y2x(−α) = −[xy(α)]9. x(−α) = x(−α) = −x(α) = −x(α);10. y(−α) = y(−α) = 2− y(α) = 2− y(α);11. xy2(−α) = xy2(−α) = −[yx(α)] = −[yx(α)];12. y2x(−α) = y2x(−α) = −[xy(α)] = −[xy(α)].Lemma 1.5 (M. Aslam Malik et al. 2005) Let n ≡ 1(mod 4). ThenQ′(√

n ) = {α ∈ Q∗(√

n ) : 2|(b, c)} and Q∗(√

n )\Q′(√

n ) = {α ∈ Q∗(√

n ) :2 - (b, c)} are both G-subsets of Q∗(

√n ).

Lemma 1.6 (M. Aslam Malik et al. 2000)

1. Let α = a+√

nc

∈ Q∗(√

n ) where n is any fixed non-square positive

integer and c is fixed. Then elements of the form a′+√

nc

and −a′+√

na′2−n

c

of

Q∗(√

n ), a′ = (a + kc), k ∈ Z, belong to αG.

2. p ≡ 1(mod 4) and α =a+√

p

q∈ Q∗

1(√

p ), for some fixed prime q,

2 < q ≤ p. Then all the elements of Q∗1(√

p ) with denominator q arealso in αG ∪ (−α)G and all the elements of Q∗

1(√

p ) with denominator−q are included in (−α)G ∪ (α)G.

3. Let p ≡ 1(mod 4) and α =a+√

p

c∈ Q∗

1(√

p ) for some c. Then all theelements of Q∗

1(√

p ) with denominator c may not be included in αG.

4

For example, if p = 37 then ±5+√

37±12

∈ (√

37 )G, −1+√

37±12

∈ (1+√

373

)G and1+√

37±12

∈ (−1+√

37−3

)G whereas these are distinct G-orbits.

It was proved in (M. Aslam Malik et al. 2000) that if p ≡ 1(mod 4),

then Q∗(√

p ) splits into at least two orbits, namely (√

p )G and (1+√

p

2)G. In

the following section, we find the ambiguous lengths of these G-orbits as afunction of p which help us in determining the number and structure of theremaining G-orbits of Q∗(

√p ).

Lemma 1.7 (Q. Mushtaq. 1998) If a circuit contains with α its conjugateα then the circuit is of the type (n1, n2, ..., nk−1, n2k, n2k, ..., n2, n1).

2 Ambiguous lengths of the G-orbits of Q∗(√

p )

We start with the following result whose proof follows by the definition offloor function and that will be used in the subsequent work and provide usa base to proceed furtherLemma 2.1 Let n be a non-square positive integer. Then

1. n = i + b√nc2 for some i ∈ N .

2. n + j = (b√nc+ 1)2 for some j ∈ N .

The following corollary is an immediate consequence of Lemma 2.1(1).Corollary 2.2 Let p ≡ 1(mod 4) such that p = a2 + b√pc2. Then:

1. p = (√

p− b√pc2 )2+b√pc2 with√

p− b√pc2 ∈ {1, 2, , . . . , b√pc−1}.

2.√

p− b√pc2 = b√pc if and only if p = 2.

3. b√pc >√

p− b√pc2 if p > 2.

4. There are some primes p ≡ 1(mod 4) which are not in the form

p = a2 + b√pc2. For example 41 = 5 + b√41c2. ¤

Lemma. 2.3 (M. Aslam Malik et al. 2004) For α ∈ Q∗(√

p ), we have thefollowings:

1. yx(α) = −α ⇔ α =1+√

p

−2or α =

−1+√

p

2

5

2. y2x(α) = −α ⇔ α =1+√

p

cor α =

−1+√

p

−c, where p = 1 + 2c.

Remarks 2.4

1. If we take√

p− b√pc2 = 1 in Corollary 2.2 (1). Then by Lemma 1.2,

the eight numbers of Q∗(√

p ) which map onto their conjugates under

x areb√pc+√p

±1,−b√pc+√p

±1,

1+√

p

±b√pc ,−1+

√p

±b√pc .

2. If we take√

p− b√pc2 = 2 in Corollary 2.2 (1). Then by Lemma 1.2,

the eight numbers of Q∗(√

p ) which map onto their conjugates under

x are2+√

p

±b√pc ,−2+

√p

±b√pc ,b√pc+√p

±2,−b√pc+√p

±2, where p > 13.

In the following results, we find the ambiguous lengths of (√

p )G and (1+√

p

2)G

of Q∗(√

p ) and show the distribution of ambiguous numbers lying in theseG-orbits.Theorem 2.5 Let p ≡ 1(mod 4) such that p − 1 is a perfect square. Thenthe circuit of (

√p )G is of the type (2b√pc, 2b√pc) and |(√p )G|amb = 8b√pc.

Proof: It is given that p− 1 = b√pc2 and hence by Lemmas 1.2 and 1.4(1)we havex(b√pc+

√p) =

−b√pc+√p

−1, x(−b√pc+

√p) =

b√pc+√p

−1,

(yx)2b√pc(−b√pc+√p) = b√pc+√p, and (xy2)2b√pc(−b√pc+√p) =b√pc+√p

−1.

Thus by Lemma 1.7 we have (y2x)2b√pc(yx)2b√pc(−b√pc+√p) = −b√pc+√p.Hence the circuit of (

√p )G is of the type (2b√pc, 2b√pc) and

|(√p )G|amb = 8b√pc which can be visualized by Figure 2.1. ¤

6

Fig.2.1. Closed path of (√

p)G where p− 1 = b√pc2.Example 2.1 By Theorem 2.5, the circuit of (

√5 )G has type (4, 4) with

|(√5 )G|amb = 16 and the circuit of (√

37 )G has type (12, 12) with |(√37 )G|amb =48. ♦

The following theorem gives the ambiguous length of (1+√

p

2)G.

Theorem 2.6 Let p > 5 be a prime such that p−1 is a perfect square. Thenthe circuit of (

1+√

p

2)G has type (1, b√pc − 1, 1, 1, b√pc − 1, 1) and hence

|(1+√

p

2)G|amb = 4(b√pc+ 1).

Proof: It is given that p − 1 = b√pc2 so Figure 2.2 can be constructed by

using Lemmas 1.2, 1.4(1) and 1.7. This follows that the circuit of (1+√

p

2)G

has type (1, b√pc−1, 1, 1, b√pc−1, 1) and hence |(1+√

p

2)G|amb = 4(b√pc+1).

¤

7

Fig.2.2. Closed path of(

(1−b√pc)+√p

2

)G

where p− 1 = b√pc2

Examples 2.2

1. By Lemma 1.1, |Q∗1(√

5 )| = τ ∗(5) = 20, so τ ∗(5) − |(√5 )G|amb = 4.

And the circuit of (1+√

52

)G has type (1, 1). Hence |(1+√

52

)G|amb = 4.Thus oG(5) = 2.

2. Let p = 37. Then by Theorem 2.5, the type of the circuit of (1+√

372

)G

is (1, 5, 1, 1, 5, 1) and |(1+√

372

)G|amb = 28. ♦

8

The following remark is an immediate consequence of Theorems 2.4 and 2.5.

Remark 2.7 Theorems 2.4 and 2.5 show that the numbersb√pc+√p

±1,−b√pc+√p

±1

are contained in (√

p)G and the numbers1+√

p

±b√pc ,−1+

√p

±b√pc are contained in

(1+√

p

2)G. ¤

The following results give the ambiguous lengths of (√

p)G and (1+√

p

2)G, in

terms of p, where p is as given in the statement of Remarks 2.3 (2).Theorem 2.8 Let p ≥ 13 and p ≡ 1(mod 4) such that p − 4 is a perfectsquare. Then(1). The circuit of (

√p )G has type(

1,b√pc−1

2, 2b√pc, b

√pc−1

2, 1, 1,

b√pc−1

2, 2b√pc, b

√pc−1

2, 1

)and |(√p )G|amb

= 4(3b√pc+ 1).

(2). The circuit of (1+√

p

2)G has type (b√pc, b√pc) and |(1+

√p

2)G|amb = 4b√pc.

Proof: The proof is analogous to that of Theorems 2.4 and 2.5.The typesof the circuits of (

√p )G, (

1+√

p

2)G can be visualized by Figures 2.3 and 2.4

respectively. ¤

9

Fig.2.3. Closed path of(−b√pc+√p

1

)G

; p ≥ 13 and p− 4 = b√pc2

10


(1−b√pc)+√p

1

)G

; p ≥ 13 and p− 4 = b√pc2

We conclude this section with the following example.Example 2.3 Let p = 229. Then by Theorem 2.7, |(√229 )G|amb = 184 and

circuit of (√

229 )G has type (1, 7, 30, 7, 1, 1, 7, 30, 7, 1). Also |(1+√

2292

)G|amb =

60 and circuit of (1+√

2292

)G has type (15, 15). ♦

11

3 Transitive G-subsets of Q∗(√

p ), p ≡ 1(mod 4)

The G-orbits namely, (√

p )G and (1+√

p

2)G of Q∗(

√p ) have been investigated

in (M. Aslam Malik et al. 2000). In Section 2, we have found the ambiguouslengths of these two G-orbits in terms of p. However Q∗(

√p ) may split into

more than two G-orbits including (√

p )G and (1+√

p

2)G.

So in this section, we investigate all the G-orbits of Q∗(√

p ) and exploresome results related to the G-orbits of Q∗(

√p ). We discuss the distribution

of the ambiguous elements of Q∗(√

p ) in the G-orbits and prove that ifp ≡ 1(mod 4) then oG(p) ≡ 0(mod 2).

Now if τ ∗(p) = |(√p )G|amb + |(1+√

p

2)G|amb, then we have oG(p) = 2. However

if τ ∗(p) > |(√p )G|amb+|(1+√

p

2)G|amb then we have the following lemma which

helps us to find the circuit of remaining G-orbits of Q∗(√

p ).Lemma 3.1 Let p ≡ 1(mod 4). Then

(α)G ∩ (α)G = ∅ for all α ∈ Q∗(√

p ) \ ((√

p )G ∪ (1+√

p

2)G).

Proof: Let p ≡ 1(mod 4). Then, we know (M. Aslam Malik 2000) that the

numbersa+√

p

±c,−a+

√p

±care contained in (

√p )G and the numbers

c+√

p

±a,−c+

√p

±a

are contained in (1+√

p

2)G. Hence by Lemma 1.3, we have

(α)G ∩ (α)G = ∅ for all α ∈ Q∗(√

p ) \ ((√

p )G ∪ (1+√

p

2)G). ¤

Lemma 3.2 Let p ≡ 1(mod 4). Then

(α)G = (−α)G and (α)G = (−α)G for all α ∈ Q∗(√

p ) \ ((√

p )G ∪ (1+√

p

2)G).

Proof: Let p ≡ 1(mod 4). Then by Lemma 3.1,

(α)G ∩ (α)G = ∅ for all α ∈ Q∗(√

p ) \ ((√

p )G ∪ (1+√

p

2)G).

Also we know that if a is a quadratic residue of p. Then −a is also a quadraticresidue of p if and only if p ≡ 1(mod 4). This completes the proof. ¤Remarks 3.3 Let n be a non-square positive integer. Then:

1. n−a2, a = odd, can be written as a product of primes in a unique wayas:

n− a2 = 2hqβ1

1 qβ2

2 . . . qβss

where 2 < q1 < q2 < . . . < qs, βi ≥ 1, 1 ≤ i ≤ s. Also if n ≡ 0(mod 2)then h = 0, if h = 1 then n ≡ 3(mod 4). Similarly h = 2 orh ≥ 3 according as n ≡ 5 or 1(mod 8).

2. n− a2, a = even, has a unique prime decomposition as:

n− a2 = 2htδ11 tδ22 . . . tδll

12

where 2 < t1 < t2 < . . . < tl, δi ≥ 1, 1 ≤ i ≤ l.Also if n ≡ 1(mod 2) then h = 0, if h = 1 then n ≡ 2(mod 4).Similarly for a2 ≡ 0(mod 8), we have h = 2 or h ≥ 3 according asn ≡ 4 or 0(mod 8) and for a2 ≡ 4(mod 8), we have h = 2 or h ≥ 3according as n ≡ 0 or 4(mod 8). ¤

Lemma 3.4 Let n ≡ 1(mod 4). Then1+√

n4

∈ Q′(√

n ) or Q∗(√

n ) \Q′(√

n ) according as n ≡ 1 or 5(mod 8).Proof: The proof of this assertion is clear from congruence relation. ¤

Lemma 3.5 Let p ≡ 1(mod 4) such that p− 1 is a perfect square.If (Q∗(

√p ) \Q′(

√p )) \ (

√p )G 6= ∅, then

either1+√

p

q1or

2+√

p

t1∈ (Q∗(

√p ) \Q′(

√p )) \ (

√p )G.

Proof Let p ≡ 1(mod 4) such that p− 1 is a perfect square. Then

by Theorem 2.4, (√

p )Gamb = {±a+

√p

±1,±a+

√p

∓(p−a2), where 0 ≤ a ≤ b√pc}.

If (Q∗(√

p )\Q′(√

p ))\(√p )G 6= ∅, then either p−1 is a power of 2 or is not a

power of 2. In the latter case, there exists1+√

p

q1∈ (Q∗(

√p )\Q′(

√p ))\(√p )G.

However if p− 1 is a power of 2 then clearly p− 4 is not a power of 2 and inthis case there exists

2+√

p

t1∈ (Q∗(

√p ) \Q′(

√p )) \ (

√p )G. ¤

Following corollary is an immediate consequence of Lemmas 3.1 and 3.5.Corollary 3.6 Let p ≡ 1(mod 4) such that p− 1 is a perfect square.If (Q∗(

√p ) \Q′(

√p )) \ (

√p )G 6= ∅ then either

(√

p )G ∪ (1+√

p

q1)G ∪ (

−1+√

p

−q1)G ⊆ Q∗(

√p ) \Q′(

√p ) or

(√

p )G ∪ (2+√

p

t1)G ∪ (

−2+√

p

−t1)G ⊆ Q∗(

√p ) \Q′(

√p ). ¤

Lemma 3.7 Let p ≡ 1(mod 8) such that p − 1 is a perfect square. ThenQ∗(

√p ) splits into at least six G-orbits for p > 17.


by Theorem 2.4, (√

p )Gamb = {±a+

√p

±1,±a+

√p

∓(p−a2), where 0 ≤ a ≤ b√pc} and by

Theorem 2.5, (1+√

p

2)Gamb = {±a+

√p

±2,±a+

√p

( p−a2

∓2),±1+

√p

±b√pc : a = 1, 3, . . . , b√pc − 1}.Also (

√p )G ⊆ Q∗(

√p ) \Q′(

√p ) and (

1+√

p

2)G ⊆ Q′(

√p ).

For p = 17, b√17c = 4 and hence ±1+√

17±b√17c ∈ (1+

√17

2)Gamb.

Thus±1+

√p

±4/∈ (

1+√

p

2)Gamb for p > 17.

Hence, for p > 17, we have always at least two more G-orbits, namely (1+√

p

4)G

and (−1+

√p

−4)G which are contained in Q′(

√p ).

13

Hence (1+√

p

2)G ∪ (

1+√

p

4)G ∪ (

−1+√

p

−4)G ⊆ Q′(

√p ).

Also by Corollary 3.6, either (√

p )G∪(1+√

p

q1)G∪(

−1+√

p

−q1)G ⊆ Q∗(

√p )\Q′(

√p )

or (√

p )G ∪ (2+√

p

t1)G ∪ (

−2+√

p

−t1)G ⊆ Q∗(

√p ) \Q′(

√p ). Hence Q∗(

√p ) splits

into at least six G-orbits for p > 17. ¤

Lemma 3.8 Let p ≡ 5(mod 8) such that p− 1 is a perfect square. Then

Q∗(√

p ) splits into at least four G-orbits namely, (√

p )G, (1+√

p

2)G, (

1+√

p

4)G

and (−1+

√p

−4)G for p > 5.


by Lemma 3.4, 1+√

n4

∈ Q∗(√

n )\Q′(√

n ). Also (√

p )G ⊆ Q∗(√

p )\Q′(√

p ).

Since for p = 5, ±1+√

5±4

∈ (√

5 )Gamb. Therefore for p > 5,

±1+√

p

±4/∈ (√

p )Gamb.

Hence (1+√

p

4)G and (

−1+√

p

−4)G always exists for p > 5 and are contained in

Q∗(√

n )\Q′(√

n ). Thus (√

p )G∪ (1+√

p

4)G∪ (

−1+√

p

−4)G ⊆ Q∗(

√p )\Q′(

√p ).

Also (1+√

p

2)G ⊆ Q′(

√p ). Hence Q∗(

√p ) splits into at least four G-orbits

namely, (√

p )G, (1+√

p

2)G, (

1+√

p

4)G and (

−1+√

p

−4)G for p > 5. ¤

To discuss the remaining G-orbits of Q∗(√

p ), p ≡ 5(mod 8), we need thefollowing results.Lemma 3.9 Let p ≡ 1(mod 4) such that p− 1 is a perfect square. Thenp ≡ 1 or 5(mod 8) according as b√pc ≡ 0 or 2(mod 4).Proof: The proof is straightforward. ¤

In particular, if b√pc = 2q1 in Lemma 3.9 then we have the followingcorollary.Corollary 3.10 Let p ≡ 1(mod 4) such that p − 1 is a perfect square andb√pc = 2q1. Then τ(p− 1) = 9.Proof:

Let b√pc = 2q1. Then p−1 = b√pc2 = (2q1)2. Hence τ(p−1) = τ(2q1)

2 =9 and these divisors of p − 1 = b√pc2 are 1, p − 1, 2, p−1

2, 4, p−1

4, b√pc, q1 =

b√pc2

, p−1b√pc

2

or 1, b√pc2, 2, b√

pc22

, 4,b√pc2

4, b√pc, q1 =

b√pc2

, 2b√pc. ¤We have seen that if p ≡ 1(mod 4) such that p − 1 is a perfect square,

then {±1+√

p

±c∈ Q∗(

√p ) : c = 1, b√pc2} ⊂ (

√p)G and {±1+

√p

±c∈ Q∗(

√p ) :

c = 2,b√pc2

2, b√pc} ⊂ (

1+√

p

2)G. Now we check the G-orbits for

1+√

p

q1and

1+√

p

4∈ Q∗(

√p ), p ≡ 5(mod 8) in the following results.

14

Theorem 3.11 Let p ≡ 5(mod 8) such that p− 1 = b√pc2 = (2q1)2. Then

the circuits of (1+√

p

4)G and (

−1+√

p

−4)G are of the type (

b√pc−2

2, 3, 1,

b√pc−2

2, 3, 1)

and hence |(1+√

p

4)G|amb = 2b√pc+ 12 = |(−1+

√p

−4)G|amb.

Proof: Let γ =3−b√pc+√p

4. Then (yx)

b√pc−2

2 (γ) =b√pc−1+

√p

2. This implies

(x)(yx)b√pc−2

2 (γ) =−b√pc+1+

√p

−b√pc/2. // Now (xy2)2(x)(yx)

b√pc−2

4 (γ) =1+√

p

−q1.

Then (xy2)(xy2)2(x)(yx)b√pc−2

4 (γ) =q1+1+

√p

−q1.

Also (xy)(xy2)3(x)(yx)b√pc−2

4 (γ) = −γ. By repeated application of Lemma1.4(7,8) we have

(y2x)(yx)3(y2x)b√pc−2

2 (yx)(y2x)3(yx)b√pc−2

2 (γ) = γ.


1+√

p

4

)G

; p ≥ 13

Hence |(1+√

p

4)G|amb = 2b√pc+ 12 and by Lemma 1.5,

15

|(1+√

p

4)G|amb = |(−1+

√p

−4)G|amb. ¤

From Theorem 3.11, we can immediately deduce the following remarks.Remark 3.12 Let p ≡ 5(mod 8) such that p− 1 = b√pc2 = (2q1)

2. Then

(1+√

p

4)G = (

1+√

p

q1)G and (

−1+√

p

−4)G = (

−1+√

p

−q1)G.

The following corollary is an immediate consequence of Lemma 3.8 andRemark 3.12.Corollary 3.13 Let p ≡ 5(mod 8) such that p− 1 = b√pc2 = (2q1)

2. ThenQ∗(

√p ) splits into at least four G-orbits namely,

(√

p )G, (1+√

p

2)G, (

1+√

p

4)G = (

1+√

p

q1)G and (

−1+√

p

−4)G = (

−1+√

p

−q1)G.

In particular, when p ≤ 2011 then oG(p) = 4 where p > 5. ¤To be more concrete, consider the following example.Example 3.1 Since b√101c = (2)(5). Then by Corollary 3.13 and Lemma

1.6 (2), Q∗(√

101 ) splits into at least four G-orbits namely, (√

101 )G, (1+√

1012

)G,

(1+√

1014

)G = (1+√

1015

)G and (−1+√

101−4

)G = (−1+√

101−5

)G. By Theorems 2.4, 2.5

and 3.11, |(√101 )G|amb = 80, |(1+√

1012

)G|amb = 44, |(1+√

1014

)G|amb = 32 =

|(−1+√

101−4

)G|amb. By Lemma 1.1, |Q∗1(√

101 )| = τ ∗(101) = 188. Since

τ ∗(101) = |(√101 )G|amb + |(1+√

1012

)G|amb + |(1+√

1013

)G|amb + |(−1+√

1013

)G|amb.Hence oG(101) = 4 ♦Examples 3.2

1. Let p = 37 ≡ 1(mod 4). We explore the G-orbits of Q∗(√

37 ) in thefollowing manner.Step-I: First we write 37 − a2, 1 ≤ a ≤ b√37c into its prime decom-position in order to find the positive divisors of 37− a2:37− 1 = 36 = 22.32, 37− 4 = 33 = 3.11, 37− 9 = 28 = 22.7,37− 16 = 21 = 3.7, 37− 25 = 12 = 22.3 and 37− 36 = 1Step-II: By Theorem 2.4, (

√37 )G

amb = {±a+√

37±1

, ±a+√

37∓(37−a2)

, where 0 ≤a ≤ 6} and by Theorem 2.5, (1+

√37

2)Gamb = {±a+

√37

±2, ±a+

√37

( 37−a2

∓2), ±1+

√37

±6:

a = 1, 3, 5}.Step-III: A = Q∗

1(√

37 )\ ((√

37 )G∪ (1+√

372

)G) 6= ∅ and 3 is the small-

est odd prime divisor of 37−a2, 1 ≤ a ≤ 6. So we take ±1+√

37±3

∈ A and

by Lemmas 3.1 and 3.2, we get (1+√

373

)G = (1+√

37−3

)G and (−1+√

37−3

)G =

(−1+√

373

)G.Step-IV: By Lemma 1.6 (2) and by M Aslam Malik , M Asim 2011,

16

±a+√

37±3

∈ Q∗1(√

37 ) are contained in B = (1+√

373

)G ∪ (−1+√

37−3

)G where

a = 1, 2, 4, 5 and by Lemma 1.6 (1), ±a+√

37

±(c= 37−a2

3)∈ Q∗

1(√

37 ) are also

contained in B where c = 12, 11, 7, 4. Now by Lemma 1.6 (1, 2),±4+

√37

±7∈ B implies that ±3+

√37

±7and ±3+

√37

±4∈ B. Also by Lemma

1.9 (1), ±5+√

37±4

∈ B implies that ±1+√

37±4

and ±1+√

37±9

∈ B. Since

Q∗1(√

37 ) \ ((√

37 )G ∪ (1+√

372

)G ∪B) = ∅. Hence oG(37) = 4.

2. By adopting the algorithm used in Example 3.2 (1), we discuss theprime p for which oG(p) = 6.Let p = 229 ≡ 1(mod 4). Then by Theorems 2.8 and 2.9, |(√229 )G|amb =

184 and |(1+√

2292

)G|amb = 60. By Lemma 1.1, oG(229) > 2. Now by

Lemma 1.9 (2), we pick 1+√

2293

∈ T = Q∗1(√

229 ) \ ((√

229 )Gamb ∪

(1+√

2292

)Gamb). Then by Lemma 3.1, we have two more G-orbits namely,

(1+√

2293

)G and (−1+√

229−3

)G of Q∗(√

229 ). By using Lemma 1.9 (1, 2), we

see that the circuits of (1+√

2293

)G and (−1+√

229−3

)G have type (9, 1, 2, 2, 5, 1,

1, 1, 2, 9, 1, 2, 2, 5, 1, 1, 1, 2). Thus |(1+√

2293

)G|amb = 96 = |(−1+√

229−3

)G|amb.Since τ ∗(229)− (184+60+96+96) = 56. Therefore oG(229) > 4. Now

we take 1+√

2296

∈ T \((1+√

2293

)Gamb∪(−1+

√229

−3)Gamb). Then by Lemma 3.1,

two more G-orbits of Q∗(√

229 ) are (1+√

2296

)G and (−1+√

229−6

)G. Again

using Lemma 1.9 (1), the circuits of (1+√

2296

)G and (−1+√

229−6

)G have

type (2, 2, 1, 4, 2, 1, 2) and hence |(1+√

2293

)G|amb = 28 = |(−1+√

229−3

)G|amb.Since τ ∗(229) = 184+60+96+96+28+28. Hence oG(229) = 6 ♦

Following are the 21 primes p ≡ 1(mod 4) and p ≤ 2011 such thatoG(p) = 4.37, 101, 197, 269, 349, 373, 389, 557, 677, 701, 709, 757, 829, 877, 997,1213, 1301, 1613, 1861, 1949, 1973.Note that 37 is the smallest prime congruent to 1(mod 4) such thatoG(37) = 4.

4 Results and Discussions

Following 7 primes p ≡ 1(mod 4) and p ≤ 2011 are such that oG(p) = 6.229, 257, 761, 733, 1229, 1373, 1489.

17

Note that the smallest prime p ≡ 1(mod 4) such that oG(p) = 6 is 229.Summing up the above results we conclude this paper with the following

main theorem.Theorem 4.1 Let p ≡ 1(mod 4). Then the number oG(p) ≡ 0(mod 2).Proof It has been proved in (M. Aslam Malik et al. 2000) that if p ≡1(mod 4), then Q∗(

√p ) splits into at least two G-orbits, namely (

√p )G and

(1+√

p

2)G.

So B = Q∗(√

p )\((√p )G∪(1+√

p

2)G) may or may not be empty. If B = ∅, then

oG(p) = 2 . However if B 6= ∅, then by Lemma 3.1, we get two more G-orbits,namely (α)G and (α)G,for some α ∈ B. Again if B \ ((α)G ∪ (α)G) = ∅, thenoG(p) = 4, otherwise we continue this process of forming the orbits, which ateach step adds two more orbits in the previous number of orbits.Since τ ∗(p) = |Q∗

1(√

p )| is finite. So after a finite number of steps, all theambiguous numbers are exhausted in forming the circuits of these orbits ofQ∗(

√p ). It follows that the number oG(p) of all G-orbits of Q∗(

√p ) for

p ≡ 1(mod 4) is congruent to 0(mod 2). ¤Note: oG(p) = 2 for all primes p ≡ 1(mod 4) and p ≤ 2011 other than listedin Table and mentioned at the ends of Examples 3.1 and 3.2.

18

TABLE

Primes p ≡ 1(mod 4) such that p ≤ 2011, oG(p) > 6.

p G-orbits αG of Q∗(√

p ) with |αG|amb oG(p) τ ∗(p)

401 |(√p )G|amb = 160, |(1+√

p

2)G|amb = 84, |(1+

√p

4)G| = 52 10 596

|(−1+√

p

−4)G|amb = 52, |(1+

√p

5)G|amb = 48 = |(−1+

√p

−5)G|amb

|(1+√

p

8)G|amb = 36 = |(−1+

√p

−8)G|amb, |(1+

√p

16)G|amb = 40

|(−1+√

p

−16)G|amb=40

1093 |(√p )G|amb = 400, |(1+√

p

2)G|amb = 132, |(1+

√p

3)G| = 168 10 1284

|(−1+√

p

−3)G|amb = 168, |(1+

√p

6)G|amb = 52 = |(−1+

√p

−6)G|amb

|(1+√

p

7)G|amb = 120 = |(−1+

√p

−7)G|amb, |(1+

√p

14)G|amb = 36

|(−1+√

p

−14)G|amb=36

1429 |(√p )G|amb = 504, |(1+√

p

2)G|amb = 164, |(1+

√p

3)G| = 264 10 1836

|(−1+√

p

−3)G|amb = 264, |(1+

√p

6)G|amb = 84 = |(−1+

√p

−6)G|amb

|(1+√

p

7)G|amb = 184 = |(−1+

√p

−7)G|amb, |(1+

√p

14)G|amb = 52

|(−1+√

p

−14)G|amb=52

1901 |(√p )G|amb = 360, |(1+√

p

2)G|amb = 196, |(1+

√p

4)G| = 120 12 1444

|(−1+√

p

−4)G|amb = 120, |(1+

√p

5)G|amb = 120 = |(−1+

√p

−5)G|amb

|(1+√

p

10)G|amb = 76 = |(−1+

√p

−10)G|amb, |(1+

√p

19)G|amb = 72

|(−1+√

p

−19)G|amb = 72, |(1+

√p

25)G|amb = 56 = |(−1+

√p

−25)G|amb

577 |(√p )G|amb = 192, |(1+√

p

2)G|amb = 100, |(1+

√p

3)G| = 72 14 892

|(−1+√

p

−3)G|amb = 72, |(1+

√p

4)G|amb = 60 = |(−1+

√p

−4)G|amb

|(1+√

p

8)G|amb = 52 = |(−1+

√p

−8)G|amb, |(1+

√p

9)G|amb = 40

|(−1+√

p

−9)G|amb = 40, |(1+

√p

16)G|amb = 36 = |(−1+

√p

−16)G|amb

|(3+√

p

8)G|amb = 40 = |(−3+

√p

−8)G|amb

1009 |(√p )G|amb = 280, |(1+√

p

2)G|amb = 156, |(1+

√p

3)G| = 144 14 1596

|(−1+√

p

−3)G|amb = 144, |(1+

√p

4)G|amb = 108 = |(−1+

√p

−4)G|amb

|(1+√

p

8)G|amb = 100 = |(−1+

√p

−8)G|amb, |(1+

√p

7)G|amb = 88

|(−1+√

p

−7)G|amb = 88, |(1+

√p

9)G|amb = 72 = |(−1+

√p

−9)G|amb

|(1+√

p

12)G|amb = 68 = |(−1+

√p

−12)G|amb

19

p G-orbits αG of Q∗(√

p ) with |αG|amb oG(p) τ ∗(p)

1601 |(√p )G|amb = 320, |(1+√

p

2)G|amb = 164, |(1+

√p

4)G| = 92 14 1244

|(−1+√

p

−4)G|amb = 92, |(1+

√p

5)G|amb = 80 = |(−1+

√p

−5)G|amb

|(1+√

p

8)G|amb = 68 = |(−1+

√p

−8)G|amb, |(1+

√p

16)G|amb = 44

|(−1+√

p

−16)G|amb = 44, |(1+

√p

25)G|amb = 40 = |(−1+

√p

−25)G|amb

|(3+√

p

8)G|amb = 56 = |(−3+

√p

−8)G|amb

1129 |(√p )G|amb = 280, |(1+√

p

2)G|amb = 156, |(1+

√p

3)G| = 120 18 1732

|(−1+√

p

−3)G|amb = 120, |(1+

√p

4)G|amb = 100 = |(−1+

√p

−4)G|amb

|(1+√

p

6)G|amb = 76 = |(−1+

√p

−6)G|amb, |(1+

√p

8)G|amb = 96

|(−1+√

p

−8)G|amb = 96, |(1+

√p

12)G|amb = 68 = |(−1+

√p

−12)G|amb

|(1+√

p

24)G|amb = 56 = |(−1+

√p

−24)G|amb, |(2+

√p

15)G|amb = 72

|(−2+√

p

−15)G|amb = 72, |(3+

√p

14)G|amb = 60 = |(−3+

√p

−14)G|amb

1297 |(√p )G|amb = 288, |(1+√

p

2)G|amb = 148, |(1+

√p

3)G| = 104 22 1644

|(−1+√

p

−3)G|amb = 104, |(1+

√p

4)G|amb = 84 = |(−1+

√p

−4)G|amb

|(1+√

p

6)G|amb = 68 = |(−1+

√p

−6)G|amb, |(1+

√p

8)G|amb = 52

|(−1+√

p

−8)G|amb = 52, |(1+

√p

9)G|amb = 64 = |(−1+

√p

−9)G|amb

|(1+√

p

16)G|amb = 48 = |(−1+

√p

−16)G|amb, |(1+

√p

24)G|amb = 44

|(−1+√

p

−24)G|amb = 44, |(1+

√p

27)G|amb = 40 = |(−1+

√p

−27)G|amb

|(3+√

p

7)G|amb = 56 = |(−3+

√p

−7)G|amb, |(3+

√p

14)G|amb = 44

|(−3+√

p

−14)G|amb = 44

References

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Aneesa Mughal Paper 07-11-2012

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