Quadratic Forms

Ferdinand Vanmaele

July 5, 2017

1 Introduction

Question. Consider Pythagorean triples (x, y, z), that is integers for which x2 + y2 = z2. More generally, we wishto find all integers n such that x2 + y2 = n. The naive way would be to plug in all possible integer values for x

and y. Doing so results in duplicates (for example 50 = 72 + 12 = 52 + 52) and does not shed much light on theunderlying structures.

We will construct a more effective method by using quadratic forms.

Definition 1. A quadratic form is a function Q : Z× Z→ Z, (x, y) 7→ ax2 + bxy + cy2 where a, b, c ∈ Z.

Example 1. The equation x2 + y2 = n is represented by a quadratic form with a = c = 1 and b = 0.

Remark 1.

• It is sufficient to find primitive solutions of a quadratic form, i.e. i.e. pairs (x, y) where x and y are coprime.This follows by dividing with the greatest common divisor d of x and y:

ax2 + bxy + cy2 = d2(ax2

d2+ b

xy

d2+ c

y2

d2

)= d2Q

(xd,y

d

)• Changing signs is negated in the quadratic form, as Q(x, y) = Q(−x,−y).

• The case (1, 0) is not excluded: Q(1, 0) = ax2.

It follows that we can identify solutions (x, y) of Q(x, y) = n with reduced fractions xy on the Farey diagram.

2 The Topograph

Definition 2. We construct the dual graph to the Farey diagram, with following properties:

• Vertices: barycentres of each triangle;

• Edges: connect vertices such that each edge of the Farey diagram is crossed exactly once;

• Regions: the areas delimited out by the edges, adjacent to a single vertex xy of the Farey diagram. We

assign the value Q(x, y) to the region.

– As every reduced fraction is represented by a vertex on the Farey diagram, every value n of Q(x, y) isrepresented as a region on the dual graph.

We call this graph the topograph of Q(x, y). (see Figure 1)

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Figure 1: Topograph

Example 2. The quadratic form Q(x, y) = x2 + y2 has the topograph shown in Figure 2. The regions have valuesQ(1, 1) = 2, Q(1, 2) = 5, and so on.

Figure 2: Topograph for x2 + y2

Remark 2. The topograph has symmetry depending on the chosen quadratic form:1

• Horizontal symmetry, when Q(−x, y) = Q(x,−y) = Q(x, y);

• Vertical symmetry, when Q(x, y) = Q(y, x).1A more complicated form is skew symmetry, described later.

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3 Arithmetic Progression

Definition 3. An arithmetic progression is a sequence of numbers such that the difference between consecutiveterms is constant.2 That is:

an = a0 + nd

For our considerations, (an)n∈N ⊂ Z and d ∈ Z.

Proposition 1 (Arithmetic progression rule). Let p, q, r and s be the four regions surrounding any given edge inthe topograph. Then the three numbers (p, q + r, s) form an arithmetic progression.

Example 3. Let Q(x, y) = x2 + y2, with the edge in the topograph that separates the regions:

p := Q(1, 0) = 1, q := Q(1, 1) = 2, r := Q(0, 1) = 1

From the arithmetic progression (p, p+ q, s) = (1, 1 + 2 = 3, s), it follows that s = 5.

Figure 3: Arithmetic progression (regions)

Proof. (Arithmetic Progression Rule)Let f : Q → Z, x

y 7→ Q(x, y) be the evaluation of the topograph on the vertices of the Farey diagram. Let

f(

x1

y1

)= q and f

(x2

y2

)= r be two vertices with corresponding regions q and r. Then by the mediant rule for

labeling vertices, the labels on the regions p and s are given by x1−x2

y1−y2and x1+x2

y1+y2:

Figure 4: Mediant rule

(These labels are correct even for x1

y1= 1

0 and x2

y2= 0

1 ).

2https://www.encyclopediaofmath.org/index.php/Arithmetic_progression

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For a quadratic form Q(x, y) = ax2 + bxy + cy2 we then have:

s = Q(x1 + x2, y1 + y2) = a(x1 + x2)2 + b(x1 + x2)(y1 + y2) + c(y1 + y2)

2

= a(x21 + 2x1x2 + x2

2) + b(x1y1 + x1y2 + x2y1 + x2y2) + c(y21 + 2y1y2 + y22)

= ax21 + bx1y1 + cy21︸ ︷︷ ︸

Q(x1,y1)=q

+ ax22 + bx2y2 + cy22︸ ︷︷ ︸

Q(x2,y2)=r

+(2ax1x2 + b(x1y2 + x2y1) + 2cy1y2)︸ ︷︷ ︸=:R

Similarly we have:

p = Q(x1 − x2, y1 − y2) = a(x1 − x2)2 + b(x1 − x2)(y1 − y2) + c(y1 − y2)

2

= a(x21 − 2x1x2 + x2

2) + b(x1y1 − x1y2 − x2y1 + x2y2) + c(y21 − 2y1y2 + hy22)

= ax21 + bx1y1 + cy21︸ ︷︷ ︸

Q(x1,y1)=q

+ ax22 + bx2y2 + cy22︸ ︷︷ ︸

Q(x2,y2)=r

−(2ax1x2 + b(x1y2 + x2y1) + 2cy1y2)

Computing p+ s results in the canceling of the R term (which is the same for both p and s), leaving:

p+ s = 2(q + r)

This equation can be rewritten as (q+ r)− p = s− (q+ r), which just says that (p, q+ r, s) forms an arithmeticprogression.

Remark 3.

• For any given vertex in the topograph, compute the three surrounding regions by inserting the correspondingfractions on the Farey diagram. Using the proposition, we can then calculate all other regions (and with that,all other values of Q(x, y)) from these three regions only. For Q(x, y) = ax2 + bxy + cy2, an easy place tostart is Q(1, 0) = a, Q(0, 1) = c and Q(1, 1) = a+ b+ c.

• Conversely, specify any three values a, b, c around any vertex of the topograph. Consider the quadraticform:

Q(x, y) = ax2 + (b− a− c)xy + cy2

We then have:Q(1, 0) = a, Q(0, 1) = c, Q(1, 1) = a+ (b− a− c) + c = b

Example 4. Consider the quadratic form with both positive and negative values Q(x, y) = x2 − 2y2. As we willsee later, this is an example of a hyperbolic form.

• Let p = Q(1, 0) = 1, q = Q(1, 1) = −1, r = Q(0, 1) = −2. What is s?;

Proposition 1(p, q + r, s) = (1,−3, s)⇒ s = −7

• p = −1, q = −2, r = −7; (p, q + r, s) = (−1,−9, s)⇒ s = −17

• p = Q(0, 1) = −2, q = Q(1, 1) = −1, r = Q(1, 0) = 1.; (p, q + r, s) = (−2, 0, s)⇒ s = 2.

We can arbitrarily continue this for other regions of the topograph. (see Figure 5)

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Figure 5: Topograph for x2 − 2y2

4 Application to continued fractions

4.1 Periodic separator line

Definition 4. Let Q(x, y) be a quadratic form that takes on both positive and negative values. We get theseparator line if we straighten out the zigzag path of edges in the topograph that separate negative values toa line.

Figure 6: Separator line for x2 − 2y2

Example 5. We again consider Q(x, y) = x2 − 2y2. We will construct the separator line using the arithmeticprogression rule.

a) p = Q(0, 1) = −2, q = Q(1, 1) = −1, r = Q(1, 0) = 1.The horizontal line segments separates positive from negative values.

b) As in the example for the arithmetic progression rule, we get (p, q + r, s) = (−2, 0, s)⇒ s = 2.Positive, place above the separator line.

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Figure 7: Construction of the separator line

c) p = 1, q = −1, r = 2 ; (p, q + r, s) = (1, 1, s)⇒ s = 1.Positive, place above the separator line.

d) p = 2, q = −1, r = 1 ; (p, q + r, s) = (2, 0, s)⇒ s = −2.Negative, place below the separator line.

e) p = −1, q = 1, r = −2 ; (p, q + r, s) = (−1,−1, s)⇒ s = −1.Negative, place below the separator line.

As we have now returned to a), further repetitions produce a periodically repeating pattern as we move to theright. The arithmetic progression rule implies that it also repeats perodically to the left, so it is periodic in bothdirections. We have found a periodic separator line for Q(x, y) = x2 − 2y2.

Remark 4. As we move upward from the separator line, the values of Q(x, y) become larger and larger, approaching+∞ monotonically. As we move downward, the values approach −∞ monotonically. The reason for this will becomeclear when we discuss the Monotonicity Property in the next talk.

Example 6. Let Q(x, y) = x2 − y2. This is an example of a 0-hyperbolic form, i.e. a form which takes on positiveand negative values as well as 0. In particular, Q = x2 − dy2 with d a square (1 = 12). This form does not have aperiodic (but monotonic) separator line:

Figure 8: Separator line for x2 − y2

This follows from the second arithmetic progression rule, where we label the boundaries of the topograph regionswith the difference of terms in an arithmetic progression (see next talk):

Proposition 2. Q1(x, y) = x2 − 2y2 has the same positive and negative values. Q2(x, y) = x2 − 3y2 has differentpositive and negative values.

Proof. Construct the periodic separator line for Q2(x, y) in a similar way as we have done for Q1(x, y). Below theseparator line, we have the periodic pattern (−2,−3, . . .) and above it (1, 1, . . .) (see Figure 10). Continued appli-cation of the arithmetic progression rule results in the regions (−2,−3,−11,−23,−26 . . .), repeated horizontally.These sequences decrease monotonically and go towards −∞, by the Monotonicity Property.

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Figure 9: Second arithmetic progression rule

Using a similar approach above the separator line, we get have sequences (1, 6, 13, 22, . . .) that go towards +∞.This implies that Q2(x, y) has at least two different positive and negative values (for example, −2 which does notappear in the sequence 1, 6, 13, 22, . . .).

Figure 10: Separator line for x2 − 3y2

Now consider the separator line of Q1(x, y). There is a skew symmetry that moves the negative values on theseparator line to the positive values (see Figure 6). By the arithmetic progression rule, it follows that Q1(x, y) hasthe same positive and negative values.

4.2 Continued fractions

Question. Let Q(x, y) = x2 − dy2, where d is a positive integer that is not a square. Can we compute the infinitecontinued fraction for

√d using the topograph of Q(x, y)?

Notation ((Continued fractions)).

[m0 : m1,m2,m3, . . .] := m0 +1

m1 +1

m2+1

m3+···

If m1,m2,m3 repeat periodically, we write [m0 : m1,m2,m3].

Remark 5. The topograph of the form x2 − dy2 always has a periodic separator line whenever d is a positiveinteger that is not a square. We will prove this in the next talk. Since the form takes the positive value 1 on 1

0 andthe negative value −d on 0

1 , this separator line always includes the edge separating these fractions.

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Example 7. We again consider the form Q(x, y) = x2 − 2y2. Superimpose the triangles of the Farey diagramcorresponding to this part of the topograph to obtain an infinite strip of triangles.

Figure 11: Strip of triangles for [1 : 2]

This strip corresponds to the infinite fraction [1 : 2] (the triangles correspond to the cutting sequence L1R2L2 . . .)3.We will use the quadratic form Q(x, y) = x2 − 2y2 to calculate the value of the infinite fraction.

The values n on the separator line for x2 − 2y2 are either ±1 or ±2. We rewrite this equation as:4(x

y

)2

= 2 +n

y2

Moving right of the diagram, we get for x, y →∞:

limx,y→∞

(2 +

n

y2

)= lim

x,y→∞2 = 2 n ∈ {±1,±2}

⇒ limx,y→∞

x

y=√2

Thus [1 : 2] =√2. Similarly, we obtain [1 : 1, 2] =

√3 using the quadratic form x2 − 3y2 (see Figure 12).

Figure 12: Strip of triangles for [1 : 1, 2]

Example 8. Taking a closer look at the figure, we notice that it is not necessary to superimpose the abovetriangles to compute the period of a continued fraction. It suffices to count the downward and upward edges of thetopograph (instead of triangles), starting from the edge separating 1

0 and 01 .

This can be illustrated with the quadratic form Q(x, y) = x2 − 7y2 with continued fraction√7 =

[2 : 1, 1, 1, 4

](see Figure 13).

3See Proposition 1 in the talk “Continued Fractions And Cutting Sequences”.4Again noting that 1

0is a valid fraction in the Farey diagram.

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Figure 13: Separator line for x2 − 7y2

Remark 6. The periodicity implies (with similar reasoning as for x2 − 2y2) that the continued fraction for√d has

the form: √d = [a0 : a1, a2, . . . , an]

Conversely, every infinite, periodic continued fraction has a corresponding hyperbolic form (but not necessarilyx2 − dy2), as proved in the next talk.

Remark 7.

• The separator line has horizontal symmetry:

Q(x, y) = x2 − dy2 = Q(−x, y)

Furthermore, for a continued fraction [a0 : a1, a2, . . . , an], an = 2a0 and the intermediate terms a1, a2, . . . , an−1form a palindrome.

• Quadratic forms where√d has an uneven period (such as x2 − 13y2) also have a skew symmetry or an

additional glide-reflection (that is, reflection plus translation) along the strip that interchanges the positiveand negative values of the form.5

Figure 14: Separator line for x2 − 13y2

Remark 8. Squares from fractions√

pq can be computed similarly to

√d using the quadratic form qx2 − py2.

As with x2− dy2, these forms always possess a periodic separator line, assuming that p and q are not both squares.The palindrome property and the relation an = 2a0 still hold for the continued fractions of

√pq , assuming that

a0 > 0, i.e. pq > 1.

5Doubling the period corresponds to ignoring the glide-reflection, and just considering the translational periodicity.

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