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arXiv:math/0606283v2 [math.NT] 1 Feb 2007

AN ELEMENTARY PROOF OF UNIQUENESS OF MARKOFF

NUMBERS WHICH ARE PRIME POWERS

YING ZHANG

Abstract. We present a very elementary proof of the uniqueness of Markoff

numbers which are prime powers or twice prime powers, in the sense that it

uses neither algebraic number theory nor hyperbolic geometry.

1. Introduction

1.1. Markoff numbers. In his celebrated work on the minima of indefinite binary

quadratic forms, A. A. Markoff [13] was naturally led to the study of Diophantine

equation—now known as the Markoff equation

x2+ y2+ z2= 3xyz.(1)

The solution triples (x,y,z) in positive integers are called by Frobenius [9] the

Markoff triples, and the individual positive integers occur the Markoff numbers.

For convenience, we shall not distinguish a Markoff triple from its permutation

class, and when convenient, usually arrange its elements in ascending order. Fol-

lowing Cassels [5], we call the Markoff triples (1,1,1) and (1,1,2) singular, while

all the others non-singular. It is easy to show that the elements of a non-singular

Markoff triple are all distinct.

In ascending order of their maximal elements, the first 12 Markoff triples are:

(1,1,1), (1,1,2), (1,2,5), (1,5,13), (2,5,29), (1,13,34), (1,34,89),

(2,29,169), (5,13,194), (1,89,233), (5,29,433), (89,233,610);

while the first 40 Markoff numbers as recorded in [19] are:

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433,610,985,1325,1597,2897,

4181,5741,6466,7561,9077,10946,14701,28657,33461,37666,43261,

51641,62210,75025,96557,135137,195025,196418,294685,426389,

499393,514229,646018,925765.

1.2. Sketch of Markoff’s work. Let f(ξ,η) = aξ2+ bξη + cη2be a binary qua-

dratic form with real coefficients. The discriminant of f is defined as δ(f) = b2−4ac,

and the minimum m(f) of f is defined as

m(f) = inf |f(ξ,η)|,

where the infimum is taken over all pairs of integers ξ,η not both zero.

Two quadratic forms f(ξ,η) and g(ξ,η) are said to be equivalent if there exist

integers a,b,c,d such that ad − bc = ±1 and f(aξ + bη,cξ + dη) = g(ξ,η).

Date: Version 1: June 9, 2006. Version 2: January 31, 2007.

2000 Mathematics Subject Classification. 11D45; 11J06; 20H10.

Supported by a CNPq-TWAS postdoctoral fellowship and in part by NSFC grant #10671171.

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2YING ZHANG

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x′

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y′

y

x

z

z′

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29

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13

2

1

5

1

Figure 1. Tree structure of a Markoff triple and its neighbors

Then Markoff’s aforementioned work on the minima of real indefinite binary

quadratic forms can be stated as follows.

Markoff’s Theorem. Let f be a real indefinite binary quadratic form. Then

inequality m(f)/?δ(f) > 1/3 holds if and only if f is equivalent to a multiple of a

Markoff form.

Here the Markoff form associated to a Markoff triple (m,m1,m2) with m ≥

m1≥ m2is defined as an indefinite binary quadratic form with integer coefficients,

as follows. First, let u be the least non-negative integer such that

um1≡ m2

(mod m) orum1≡ −m2

(mod m).

Since 0 ≡ m(3m1m2−m) = m2

in §1.5, gcd(m1,m) = 1, we have u2+ 1 ≡ 0 (mod m). Now let

1+m2

2≡ (u2+1)m2

1(mod m) and, as will be shown

v = (u2+ 1)/m.

The Markoff form associated to Markoff triple (m,m1,m2) is then defined as

φ(m,m1,m2)(ξ,η) = mξ2+ (3m − 2u)ξη + (v − 3u)η2. (2)

Note that for φ := φ(m,m1,m2)we have δ(φ) = 9m2− 4 and m(φ) = m.

Remark. Note that Markoff [12] [13] used continued fractions to obtain his results,

and his proofs were only sketched. Dickson [8, Ch.VII] gave a detailed interpretation

of it. Frobenius [9] made a systematic study of the Markoff numbers, based on which

Remak [15] presented a proof of Markoff’s Theorem using no continued fractions.

Markoff’s above result also has a well-known equivalent formulation in terms of the

approximation of irrationals by rationals; see Cassels [5] and Cusick–Flahive [7] for

detailed explanations.

1.3. Neighbors of a Markoff triple. That Markoff equation (1) is particularly

interesting lies in the fact that it is a quadratic equation in each of the variables,

and hence new solutions can be obtained by a simple process from a given one,

(x,y,z). To see this, keep x and y fixed and let z′be the other root of (1), regarded

as a quadratic equation in z. Rewriting (1) as z2− 3xyz + (x2+ y2) = 0, we have

z+z′= 3xy and zz′= x2+y2. Thus z′is a positive integer and (x,y,z′) is another

solution triple to (1) in positive integers, that is, a Markoff triple. Similarly, we

obtain two other Markoff triples (x′,y,z) and (x,y′,z). We call these three new

Markoff triples thus obtained the neighbors of the (x,y,z). See Figure 1 for an

illustration.

1.4. Reduction. In [13, pp.397–398], Markoff showed that every Markoff triple

can be obtained from the simplest by appropriately iterating the above process.

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UNIQUENESS OF MARKOFF NUMBERS WHICH ARE PRIME POWERS3

The Reduction Theorem. Every Markoff triple can be traced back to (1,1,1)

by repeatedly performing the following operation on Markoff triples:

(x,y,z) ?−→ (x,y,z′) := (x,y,3xy − z), (3)

where the elements of (x,y,z) is arranged so that x ≤ y ≤ z.

Note that to perform the next operation, one needs to first rearrangethe elements

of (x,y,z′) in ascending order. As an example, we see

(13,194,7561) ?−→ (13,194,5) ∼ (5,13,194) ?−→ (5,13,1) ∼ (1,5,13)

?−→ (1,5,2) ∼ (1,2,5) ?−→ (1,2,1) ∼ (1,1,2) ?−→ (1,1,1).

A simple proof of the theorem is given in [5, pp.27–28]; see also [7, pp.17–18]. The

idea is that operation (3) reduces the maximal elements of Markoff triples as long

as the input triple is non-singular. Indeed, one has x < y < z and (z −y)(z′−y) =

zz′− (z + z′)y + y2= x2+ 2y2− 3xy2< 0; hence z′< y.

Here we give a slightly different proof, the idea of which we get from [2].

Proof.

Markoff triples exactly when z′< z. Therefore, after a finite number of times of

length reduction, one stops when z′≥ z, or equivalently, 2z ≤ 3xy. We claim that

z = 1 in this case, and hence (x,y,z) = (1,1,1). Indeed, if z ≥ 2 then one obtains

from x ≤ y ≤ z and 2z ≤ 3xy that

x

3yz+

This forces that x = y,z = 2 and x = 1,y = z both hold, a contradiction.

The operation (x,y,z) ?−→ (x,y,z′) reduces the lengths, x + y + z, of

1 =

y

3zx+

z

3xy≤1

6+1 3+12= 1. (4)

?

1.5. First properties of Markoff numbers. As an immediate corollary of the

Reduction Theorem, we see that the elements of a Markoff triple are pairwise

coprime. Moreover, since zz′= x2+y2and gcd(x,y) = 1, a Markoff number is not

a multiple of 4, and each odd prime factor of a Markoff number is ≡ 1(mod 4).

Consequently, every odd Markoff number is ≡ 1(mod 4) and every even Markoff

number is ≡ 2(mod 8). Indeed, it is shown in [20] that every even Markoff number

is ≡ 2(mod 32).

1.6. An illustration. The Reduction Theorem tells that, starting from (1,2,5)

and generating new neighbors repeatedly, one will obtain all the Markoff triples.

This is depicted as an infinite binary tree in Figure 2 in which all the Markoff

numbers appear in the regions while all non-singular Markoff triples appear around

vertices. In this shape it seems to be first drawn by Thomas E. Ace on his web-page

http://www.minortriad.com/markoff.html.

1.7. The uniqueness problem. A problem then arises naturally: Does every

Markoff number appear exactly once in the regions in Figure 2? In other words,

are there any repetitions among all the numbers occur?

The following conjecture on the uniqueness of Markoff numbers/triples was first

mentioned explicitly by G. Frobenius as a question in his 1913 paper [9]. It asserts

that a Markoff triple is uniquely determined by its maximal element. (And we shall

simply say that a Markoff number z is unique if the following is true for z.)

The Unicity Conjecture.

with x ≤ y ≤ z and ˜ x ≤ ˜ y ≤ z. Then x = ˜ x and y = ˜ y.

Suppose (x,y,z) and (˜ x, ˜ y,z) are Markoff triples

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13

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34

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433

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14701

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7561

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6466

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

985

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1325

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37666

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2897

1

2

5

29

169

194

5741

499393

7453378

1278818

3276509

48928105

8399329

96557

43261

1686049

4400489

294685

51641

135137

9077

233

Figure 2. Markoff numbers in an infinite binary tree

The conjecture has been proved only for some special subsets of the Markoff

numbers. The following affirmative result for Markoff numbers which are prime

powers or twice prime powers was first proved independently and partly by Baragar

[1], Button [3] and Schmutz [17] using either algebraic number theory ([1],[3]) or

hyperbolic geometry ([17]). And a stronger result along the same lines has been

obtained later by Button in [4]; in particular, a Markoff number is shown to be

unique if it is a “small” (≤ 1035) multiple of a prime power.

Theorem 1 (Baragar [1]; Button [3]; Schmutz [17]). A Markoff number is unique

if it is either a prime power or twice a prime power.

Page 5

UNIQUENESS OF MARKOFF NUMBERS WHICH ARE PRIME POWERS5

This paper is motivated by a simple proof of Theorem 1 recently published by

Lang and Tan [11], which uses some elementary facts from the hyperbolic geometry

of the modular torus with one cusp, as used by Cohn in [6]. The aim of this paper

is to present in detail a completely elementary proof of Theorem 1 that uses neither

algebraic number theory nor hyperbolic geometry so that an average reader will be

able to fully understand it with no difficulty. Though it is later clear that all the

needed ingredients of the proof were already known as early as 1913 in Frobenius’

work, we must admit that we first obtain them from hyperbolic geometry, especially,

that used in [11] and [6].

The rest of the paper is organized as follows. In §2 we parametrize Markoff

numbers using non-negative rationals (slopes) t ∈ˆQ ∩ [0,∞]. We also define ut

as in §1.2 and verify some properties of the pairs (mt,ut). Then in §3, with the

help of a simple lemma (Lemma 4), we give the promised elementary proof of

Theorem 1. In §4 we introduce the so-called Markoff matrices to generate all

Markoff numbers. Certain properties of these matrices are then discussed in §5. In

particular, alternative proofs of Lemmas 2 and 3 will be given. Finally, in §6 we

give a geometric explanation of the Markoff numbers and related numbers.

Acknowledgements. The author would like to thank Ser Peow Tan for helpful

conversations and suggestions. Thanks are also due to a referee of the first version

of this note, whose constructive suggestions helped improve the exposition of the

current version.

2. Slopes of Markoff numbers

2.1. Slopes of Markoff numbers. It is natural and very useful to associate to

each Markoff number a slope, that is, an ordered pair of non-negative coprime

integers. This was first done by Frobenius in [9] where he set

m(1,0) = 1, m(0,1) = 2, m(1,1) = 5, m(1,2) = 13, m(2,1) = 29, ...

(These pairs are also called by Cusick and Flahive [7] the Frobenius coordinates

of Markoff numbers). Note that, by identifying (µ,ν) with ν/µ, the slopes are

nothing but the positive rationals together with 0 and ∞. In the latter context we

shall write mrfor m(µ,ν) where r = ν/µ.

Let us writeˆQ := Q ∪ {∞}. We shall also call ∞ =1

the set of slopes we consider is the set of rationals in [0,∞], that is,ˆQ ∩ [0,∞].

0=−1

0a rational. Then

2.2. Farey sum of rationals. There is a simple but useful way to obtain all

the positive rationals by making the so-called Farey sums repeatedly. Specifically,

starting with 0 =0

0(of level 0), all positive rationals can be generated,

level by level, as follows:

1and ∞ =1

1

1=0+1

1+1,2

2+1,3

3+1,4

1+0;

1=1+1

2=1+2

3=1+3

1

2=0+1

3=1+1

4=2+1

1+0;

1+1,3

1+2,5

1

3=0+1

5=1+2

1+2,2

2+3,3

1=2+1

3=3+2

1+0;

2+1,5

1

4=0+1

1+3,2

5=1+1

3+2,3

2=2+3

1+1,4

1=3+1

1+0;

and so on. (To obtain the negative rationals, one starts from ∞ =−1

instead and makes the Farey sums recursively as above). In particular, we have the

notion of Farey level for positive rationals, with levels 1 to 4 shown as above. To

0and 0 =0

1

Page 6

6YING ZHANG

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .

t

s

r′

s′

Figure 3. Farey triples (r,t,s), (r,s′,t) and (t,r′,s)

obtain all the rationals in [0,∞] of Farey level n+1, we simply start with all those

of Farey level not exceeding n, arrange them in ascending order, and make Farey

sum for each pair of adjacent ones among them. In particular, we are allowed to

prove a proposition concerning all the positive rationals by induction on the Farey

levels of the rationals involved. In what follows we shall make the above idea precise

and present some basic facts that will be needed in later part of this paper.

By the standard reduced form of a rational number t we mean the unique frac-

tional expression t = ν/µ where µ,ν are coprime integers with µ ≥ 0. Two rationals

r,s are said to be Farey neighbors (and that they form a Farey pair) if they have

standard reduced forms r = b/a and s = d/c so that ad − bc = ±1. Given a Farey

pair r,s with standard reduced forms r = b/a and s = d/c, their Farey sum is

defined as

r ⊕ s :=b + d

a + c

(5)

which is certainly in its standard reduced form. (Note that in terms of (a,b) and

(c,d) regarded as plane vectors, the Farey sum is just the vector sum). Clearly,

r ⊕s = s⊕r. It is easy to see that r ⊕s falls in between r and s and is a common

Farey neighbor of r and s. We shall call the ordered triple (r,t,s) a Farey triple.

It follows from the Euclidean algorithm that every positive rational can be writ-

ten in a unique way as the Farey sum of a Farey pair of rationals in [0,∞]. Indeed,

for a given positive rational t, among all its Farey neighbors there are exactly two,

r and s, having smaller or the same denominators; it can be easily shown that r

and s form a Farey pair and t = r ⊕ s. We call r and s the direct descents of t. As

a consequence, it is easy to see that in every Farey pair inˆQ∩[0,∞], the one with

smaller denominator or numerator has smaller Farey level and is a direct descent

of the other. Hence it can be shown by induction that all rationals between0

1

0will appear in the above process of recursively making Farey sums.

To end this subsection, we give a formal definition of the notion of Farey level.

First, we set the Farey level of each of

1and

pair r,s inˆQ ∩ [0,∞], we define the Farey level of their Farey sum t = r ⊕ s to

be the sum of their respective Farey levels. In this way we then have recursively

defined a Farey level for each t ∈ˆQ ∩ [0,∞].

1and

0

1

0to be 0. Recursively, for a Farey

2.3. Further properties of Markoff numbers. It is easy to see that, in terms

of slopes, each Markoff triple is then of the (more natural) form (mr,mt,ms) where

(r,t,s) is a Farey triple inˆQ ∩ [0,∞] with r < t < s.

Page 7

UNIQUENESS OF MARKOFF NUMBERS WHICH ARE PRIME POWERS7

We define for each t ∈ˆQ ∩ [0,∞] an integer ut with 0 ≤ ut ≤ mt as follows.

First, we set u0/1= 0, u1/0= 1. In general, for t ∈ Q ∩ (0,∞), utis defined by

ut≡ ms/mr(mod mt). (6)

Then utdepends only on t but not the triple (r,t,s) since for the neighboring Farey

triples (r,s′,t) and (t,r′,s) as shown in Figure 3 we have

mr/ms′ ≡ ms/mr≡ mr′/ms(mod mt) (7)

which in turn follows from

msms′ = m2

r+ m2

t

andmrmr′ = m2

t+ m2

s.

Now since 0 ≡ m2

from (6)

r+ m2

s≡ m2

r(1 + u2

t) (mod mt) and gcd(mr,mt) = 1, we have

u2

t+ 1 ≡ 0 (mod mt). (8)

Furthermore, we have

Lemma 2. The ratio ut/mtis strictly increasing with respect to t ∈ˆQ∩[0,∞]. In

particular, 0 ≤ ut≤ mt/2, with strict inequalities for t ?=0

1,1

0.

In fact, Lemma 2 follows from the following

Lemma 3. For every Farey triple (r,t,s) inˆQ ∩ [0,∞] with r < t < s,

ut

mt

mr

mrmt

which are equivalent respectively to

−ur

=

ms

and

us

ms

−ut

mt

=

mr

mtms, (9)

utmr− urmt= ms

andusmt− utms= mr.(10)

Proof.

The conclusion is easily checked to be true for the Farey triple (0

suppose that (10) holds for all Farey triples (r,t,s) inˆQ ∩ [0,∞] with r < t < s

and with the Farey level of t not exceeding n ≥ 1. In particular, this implies that

0 ≤ ur/mr< ut/mt< us/ms≤ 1/2.

Then we only need to show that (10) also holds for the Farey triples (r,s′,t)

and (t,r′,s) as shown in Figure 3. Since the proofs for the two cases are entirely

similar, we prove it for the case (r,s′,t) only, that is, we show that

We prove it by induction on the Farey levels of the rationals involved.

1,1

1,1

0). Now

us′mr− urms′ = mt

andutms′ − us′mt= mr.(11)

For this, we first see from (10) and m2

r+ m2

=utms′ − mr

t= ms′msthat

u :=mt+ urms′

mr

mt

.(12)

Note that 0 < u/ms′ < ut/mt< 1/2 and, by (7), u is an integer. Hence (11) holds

with us′ replaced by u. But this in turn implies that u ≡ mt/mr(mod ms′), and

hence u = us′ by the definition of us′. This proves Lemma 3.

?

Remark. The inequalities in (10) first appeared in [9, p.602], though they were

contained essentially but implicitly in [13]. The result of Lemma 2 was stated and

proved by Remak in [15]. In later part of this paper (see §5.2), Lemma 3 will also

be obtained in a nice way as a corollary of the properties (see Proposition 7) of the

so-called Markoff matrices which are interesting in their own right.

Page 8

8 YING ZHANG

2.4. Slope form of the Unicity Conjecture. In terms of slopes, we may rephrase

the Unicity Conjecture as

The Unicity Conjecture (Slope form).

ˆQ ∩ [0,∞] are all distinct.

The Markoff numbers mt, t ∈

3. Proof of Theorem 1

We are now ready to give a very elementary proof for Theorem 1, using Lemma 2

and the following simple lemma whose proof can be found in [20].

Lemma 4. Suppose m = pnor 2pnfor an odd prime p and an integer n ≥ 1.

Then, for any integer l coprime to m, the binomial congruence equation

x2+ l ≡ 0 (mod m) (13)

has at most one integer solution x with 0 < x < m/2.

Proof of Theorem 1. Suppose there exist slopes t,t∗∈ˆQ ∩ [0,∞] such that

mt= mt∗ = pnor 2pn

for an odd prime p and an integer n ≥ 1. By (8) and its analog for ut∗, Lemma 4

applies to give ut= ut∗. Then t = t∗by Lemma 2. This proves Theorem 1.

?

Remark. The reader who is interested in merely the proof of Theorem 1 may well

exit here. The rest of this paper is devoted to a discussion of the so-called Markoff

matrices, which (with the exception of §5.2) constitutes the main body of an earlier

version of this paper and can be used to prove our earlier results in a nice way.

4. Markoff matrices

It is Harvey Cohn [6] who first noticed the relationship of Markoff equation (1) and

one of Fricke’s trace identities, (16) below, for matrices in SL(2,C). This gives us

a nice way to generate the Markoff numbers using the so-called Markoff matrices

and hence to reformulate the Unicity Conjecture.

4.1. Fricke’s Trace identities. In this subsection we derive some of Fricke’s trace

identities as needed.

Proposition 5. If X,Y ∈ SL(2,C) then

tr(XY ) + tr(XY−1) = tr(X)tr(Y );(14)

tr2(X) + tr2(Y ) + tr2(XY ) − tr(X)tr(Y )tr(XY ) = 2 + tr(XY X−1Y−1).

In particular, if X,Y ∈ SL(2,C) satisfy tr(XY X−1Y−1) = −2 then

tr2(X) + tr2(Y ) + tr2(XY ) = tr(X)tr(Y )tr(XY ).

(15)

(16)

Proof.

Here, however, we include a simpler derivation as presented in, for instance, [10]

(see also [14]), which not only enables us to avoid tedious calculations but also

would led us to the rediscovery of the identities.

?a

and Y +Y−1= tr(Y )I, where I denotes the identity matrix. Then left-multiplying

the latter equality by X gives

XY + XY−1= Xtr(Y ).

These identities can be verified easily by straightforward calculations.

First, note that if Y =

b

cd

?

then Y−1=

?

d

−b

−ca

?

. Hence tr(Y ) = tr(Y−1)

(17)

Page 9

UNIQUENESS OF MARKOFF NUMBERS WHICH ARE PRIME POWERS9

Taking traces on both sides of (17), we obtain identity (14). As a special case, we

take X = Y in (14) to get

tr(Y2) = tr2(Y ) − 2. (18)

Finally, by making use of identity (14) repeatedly, we can calculate tr(XY X−1Y−1)

and thus obtain (15) easily as follows:

tr(XY X−1Y−1) = tr(X)tr(Y X−1Y−1) − tr(XY XY−1)

= tr2(X) −?tr(XY )tr(XY−1) − tr(XY Y X−1)?

= tr2(X) − tr(XY )?tr(X)tr(Y ) − tr(XY )?+ tr(Y2)

= tr2(X) − tr(X)tr(Y )tr(XY ) + tr2(XY ) + tr2(Y ) − 2.

This proves Proposition 5.

?

Remark. Many other trace identities of Fricke for matrices in SL(2,C), though

shall not be needed in this paper, have been explored in [10] in details.

4.2. Markoff matrices. Following Cohn [6] but with a different choice, we asso-

ciate a matrix in SL(2,Z) to each slope t ∈ˆQ ∩ [0,∞] as follows. Initially, we

set

?2

11

A =

1

?

,B =

?11

21

?

(19)

and define

M 0

1= A =

?21

11

?

,M 1

0= AB =

?34

32

?

. (20)

In general, for a Farey pair r,s ∈ˆQ ∪ [0,∞] with r < s, we set

Mr⊕s= MrMs(?= MsMr). (21)

Thus we have defined for every t ∈ˆQ ∪ [0,∞] a Markoff matrix, Mt∈ SL(2,Z),

with positive elements. As a few more examples, one finds

M 1

2=

?21

13

29

18

?

,M 1

1=

?8 11

75

?

,M 2

1=

?46

29

65

41

?

;

M 1

3=

?55

34

76

47

?

, M 2

3=

?313

194

434

269

?

, M 3

2=

?687

433

971

612

?

, M 3

1=

?268

169

379

239

?

.

It is easy to observe that the trace of Mtequals 3 times the (2,1)-element; for

proof, see Proposition 7(iii), §5. Thus we may write for t ∈ˆQ ∩ [0,∞]

mt:= tr(Mt)/3.(22)

Recall from §2.2 that by a Farey triple (r,t,s) inˆQ ∩ [0,∞] with r < t < s we

mean that r,s ∈ˆQ ∩ [0,∞] are a Farey pair and that t = r ⊕ s.

Proposition 6. For every Farey triple (r,t,s) inˆQ ∩ [0,∞], (mr,mt,ms) is a

Markoff triple.

Proof. This follows from a simple application of identity (16) with X = Mrand

Y = Ms. To apply (16), we need to verify that

tr(MrMsM−1

r M−1

s ) = −2 (23)

Page 10

10 YING ZHANG

for every pair of Farey neighbor r,s ∈ˆQ ∩ [0,∞] with r < s. Indeed, since

tr(MrMtM−1

it suffices to check (23) for the initial pair (r,s) = (0

r M−1

t

) = tr(MrMsM−1

r M−1

s ) = tr(MtMsM−1

t

M−1

s ),

1,1

?

0). This is true because

tr

?

M 0

1M 1

0M−1

0

1

M−1

1

0

?

= tr

?−7

−6

6

5

= −2.

Since trMr= 3mretc., we obtain from (16) that

(3mr)2+ (3ms)2+ (3mt)2= (3mr)(3ms)(3mt).

This shows that (mr,mt,ms) is a Markoff triple.

?

4.3. Matrix form of the Unicity Conjecture. In terms of Markoff matrices

defined above, we may rephrase the Unicity Conjecture as:

The Unicity Conjecture (Matrix form). The traces of Markoff matrices

Mr, r ∈ˆQ ∩ [0,∞] are all distinct.

5. Properties of Markoff matrices

The Markoff matrices defined in §4 possess certain nice properties which can be

easily observed by inspecting just a few examples.

5.1. Elements of a single Markoff matrix. In a Markoff matrix, we have

Proposition 7. For t ∈ˆQ∩[0,∞], let Mt=

?ab

cd

?

be the Markoff matrix defined

above. Then (i) c ≤ d ≤ a ≤ b; (ii) 3a ≥ 2b, 3c ≥ 2d; and (iii) a + d = 3c.

Moreover, the inequalities in (i) and (ii) are all strict when t ?= 0,∞.

Proof. We prove (i)–(iii) by induction on the Farey level of t. The conclusions

(i)–(iii) are readily seen to be true for t ∈ˆQ∩[0,∞] of Farey level up to 1, that is,

for r =0

out in §2.2, there exists a unique Farey pair r,s ∈ˆQ ∩ [0,∞] with r < s, such that

t = r ⊕ s. In particular, r and s have smaller Farey levels. Let

1,1

1,1

0. Now suppose t ∈ˆQ ∩ [0,∞] has Farey level at east 2. As pointed

Mr=

?ab

dc

?

,Ms=

?xy

wz

?

.(24)

Then, by definition,

Mt= MrMs=

?ax + bz

cx + dz

ay + bw

cy + dw

?

.(25)

To complete the inductive step, we proceed to prove (ii), (iii) and (i) in this order.

Proof of (ii) for the inductive step: It suffices to observe that

y

x<ay + bw

ax + bz<cy + dw

cx + dz<wz≤3

2.(26)

Proof of (iii) for the inductive step: We need to show

(ax + bz) + (cy + dw) = 3(cx + dz).(27)

The inductive hypothesis gives

a + d = 3c,x + w = 3z.(28)

Page 11

UNIQUENESS OF MARKOFF NUMBERS WHICH ARE PRIME POWERS11

Thus (27) is equivalent to

2dx = bz + cy. (29)

There are two possibilities: the denominator (or numerator) of r is less or greater

than that of s. Accordingly, we have s = r ⊕ t′or r = t′⊕ s, where t′∈ˆQ ∩ [0,∞]

has Farey level lower than the maximum of those of r and s. In the case where

s = r ⊕ t′we have

Mt′ = M−1

r Ms=

?d

−c

−b

a

??xy

wz

?

=

?dx − bz

−cx + az

dy − bw

−cy + aw

?

. (30)

Now the inductive hypothesis on Mt′ yields

(dx − bz) + (−cy + aw) = 3(−cx + az) (31)

which is, by (28), equivalent to (29). The proof for the other case is entirely similar.

Proof of (i) for the inductive step: The first and the last of the three

inequalities in (i), that is,

ax + bz < ay + bwand cx + dz < cy + dw

follow easily from the inductive hypothesis x ≤ y, z ≤ w, of which at least one

inequality is strict. It remains to prove

ax + bz > cy + dw. (32)

By (27), this is equivalent to

3(cx + dz) > 2(cy + dw)

which is true since we have from the inductive hypothesis that 3x ≥ 2y and 3z ≥ 2w,

and at least one of these two inequalities is strict.

This finishes the inductive step as well as the proof of Proposition 7.

?

5.2. Alternative proof of Lemma 3. By Proposition 7, every Markoff matrix

Mt, t ∈ˆQ ∩ [0,∞] is of the form

Mt=

?2mt− u

mt

∗

mt+ u

?

,

where 0 ≤ u ≤ mt/2. Now detMt= 1 implies that u2+ 1 ≡ 0 (mod mt). By the

definition of ut, we have u = ut. Thus we obtain

Proposition 8. For every t ∈ˆQ ∩ [0,∞], the Markoff matrix

Mt=

?2mt− ut

mt

2mt+ ut− vt

mt+ ut

?

.(33)

Using (33), we can give an alternative proof of Lemma 3.

Alternative Proof of Lemma 3. We obtain from MrMs= Mtthat

Ms= M−1

r Mt=

?

∗∗

∗utmr− urmt

?

.

Equating the (2,1)-elements then gives the first equality in (10). The second equal-

ity in (10) follows similarly from Mr= MtM−1

s .

?

Page 12

12YING ZHANG

5.3. Monotonicity of the index of a Markoff matrix. It is convenient to

introduce an index

̺(Mt) :=a

c

(34)

for every Markoff matrix Mt=

We then have the following monotonicity of the index of a Markoff matrix with

respect to its slope. This follows readily from Lemma 2 and (33). However, we

choose to include a direct simple proof which in turn gives an alternative proof of

Lemma 2.

?ab

cd

?

, where r ∈ˆQ ∩ [0,∞].

Proposition 9. Suppose t1,t2∈ˆQ∩[0,∞] where t1< t2. Then ̺(Mt1) > ̺(Mt2).

Proof.

the Farey levels of t1 and t2. First, the conclusion is true for t1,t2 both having

Farey level 0, that is, t1=0

By the process of constructing all the rationals in [0,∞] by recursively making

Farey sums as described in §2.2, we only need to prove the conclusion locally, that

is, suppose it is true for a Farey pair r,s ∈ˆQ ∩ [0,∞] with r < s and show

We proceed to prove this proposition by induction on the maximum of

1and t2=1

0, since ρ(M 0

1) = 2/1 and ρ(M 1

0) = 3/2.

̺(Mr) > ̺(Mt) > ̺(Ms), (35)

where we have written t := r ⊕ s. Suppose Mr, Msare given by (24). Then Mt

is given by (25). By our inductive hypothesis, ̺(Mr) > ̺(Ms), that is, a/c > x/z.

Then (35) is equivalent to

a

c>ax + bz

cx + dz>x z.

(36)

The first inequality in (36) follows easily from the fact that a/c > b/d (since

ad − bc = 1). The second is equivalent, by Proposition 7 (iii), to the inequality

cy + dw

cx + dz<w z,

which is true since y/x < w/z (as xw − yz = 1). This finishes the inductive step

as well as the proof of Proposition 9.

?

As an immediate corollary of Proposition 9, we obtain that

Proposition 10. The Markoff matrices Mt, t ∈ˆQ ∩ [0,∞] are all distinct.

Remark. There are other choices in the definition of Markoff matrices, such as

M 0

1=

?21

11

?

,M 1

0=

?52

12

?

. (37)

For this choice we have for all t ∈ˆQ ∩ [0,∞]

Mt=

?

2mt+ ut

2mt− ut− vt

mt

mt− ut

?

. (38)

6. Further remarks

In this section we make further remarks to give a geometric explanation of the

Markoff numbers and related numbers.

Page 13

UNIQUENESS OF MARKOFF NUMBERS WHICH ARE PRIME POWERS13

6.1. Once-cusped hyperbolic torus. It is known from Cohn’s work [6] that the

Markoff numbers correspond to the simple closed geodesics on a special hyperbolic

torus with a single cusp. (See [18] an exposition of the background.) Specifically,

let A,B ∈ SL(2,R) be given as in §4.2 and let ?A,B? ⊂ SL(2,R) be the subgroup

generated by A and B. Then ?A,B? is a Fuchsian group and T := H2/?A,B?

is once-cusped hyperbolic torus, where H2is the upper half-plane model of the

hyperbolic plane. Note that the axes of the M¨ obius transformations A, B and AB

project onto simple closed curves on T. Assign to these three simple closed curves

on T the slopes0

0respectively, and consider all the simple closed cures

γton T of slopes t ∈ˆQ ∩ [0,∞]. Let the hyperbolic length of γtbe lt. Then we

have the relation

3mt= 2cosh(lt/2).

1,−1

1and1

Hence the Unicity Conjecture is actually a conjecture about the uniqueness of

lengths of certain simple closed geodesics on the specific hyperbolic torus T.

6.2. McShane identity. For a Farey triple (r,t,s) inˆQ ∩ [0,∞] with r < t < s,

the quantity

mrms

meanings. In particular, it leads naturally to the interesting McShane identity; see

[2] (Theorem 3 and Proposition 3.13 there) for details.

mt′

(where mt′ = 3mrmr−mt) appeared in (9) has nice geometric

6.3. Exceptional vector bundles on CP2. In an unexpected way, the Markoff

numbers also appear as the ranks of the exceptional vector bundles on CP2, as

explained by Rudakov [16]. In this context, the quantity u/m is the “slopes” the

corresponding vector bundles, with u the first Chern class (which is an integer in

this case).

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ofcertain Markoff numbers, preprint,

Department of Mathematics, Yangzhou University, Yangzhou 225002, CHINA

E-mail address: yingzhang@yzu.edu.cn

Current address: IMPA, Estrada Dona Castorina 110, 22460 Rio de Janeiro, BRAZIL

E-mail address: yiing@impa.br

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