# The Generalizations of the Golden Ratio: Their Powers

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The Generalizations of the Golden Ratio: Their Powers, Continued Fractions, and

Convergents

Saarik Kalia

December 23, 2011

Abstract The relationship between the golden ratio and continued fractions is commonly known about throughout the mathematical world: the convergents of the continued fraction are the ratios of consecutive Fibonacci numbers. The continued fractions for the powers of the golden ratio also exhibit an interesting relationship with the Lucas numbers. In this paper, we study the silver means and introduce the bronze means, which are generalizations of the golden ratio. We correspondingly introduce the silver and bronze Fibonacci and Lucas numbers, and we prove the relationship between the convergents of the continued fractions of the powers of the silver and bronze means and the silver and bronze Fibonacci and Lucas numbers. We further generalize this to the Lucas constants, a two-parameter generalization of the golden ratio.

1 Introduction

The golden ratio exhibits an interesting relationship with continued fractions. We can summarize this relationship in three already known properties. Firstly, the nth convergent of the golden ratio is Fn+1 .[1] Secondly, the con-

Fn

tinued fraction of the nth power of the golden ratio is {Ln; Ln} for odd n and {Ln − 1; 1, Ln − 2} for even n.[3] Finally, the convergents of the powers of the

1

golden ratio can be expressed as Fa(n+1) . In this paper, we will generalize the Fan

golden ratio to a group of constants and see that corresponding properties hold.

In Section 2, we will deﬁne old and new terms and list known properties. In Section 3, we will prove the three properties, which constitute the relationship between the golden ratio and continued fractions. In Section 4, we generalize this to the silver means, whose continued fractions are {m; m}, analogous to the golden ratio’s {1; 1}. To complete the generalization, not only must we generalize the Fibonacci and Lucas numbers to two families of series we will call the silver Fibonacci and Lucas numbers, but we need to also deﬁne the bronze means and the associated bronze Fibonacci and Lucas numbers. We will ﬁnd that the three properties which were true of the golden ratio also hold for the silver and bronze means. In Section 5, we will go even further to a two-parameter generalization of these properties. The Lucas sequences are a family of sequences, of which the silver and bronze Fibonacci and Lucas numbers are a subset. When we deﬁne the Lucas constants, constants associated with a Lucas sequence and analogous to the silver and bronze means, we see that they obey one of the properties. The other two, however, only certain Lucas constants obey. We will discuss the nature of the continued fractions of the Lucas constants and for which Lucas constants these properties hold. Though we will ﬁnd certain cases for which these properties hold, the search for an all-encompassing case will be left to further research.

2 Deﬁnitions and Properties

2.1 Old Deﬁnitions

A continued fraction is a form of representing a number by nested fractions,

all of whose numerators are 1. For instance, the continued fraction for 79

1

is 1 +

. The compact notation for this continued fraction is {1; 3, 2}.

1

3+

2

(Note a semicolon follows the ﬁrst term, while commas follow the others.)

The continued fraction of a number is ﬁnite if and only if that number is

rational.

2

A quadratic irrational or quadratic surd is a number that is the solution to some quadratic equation with rational coeﬃcients. The continued fraction of a number is periodic, meaning it has a repeating block, if and only if that number is a quadratic irrational. The repeating block of a periodic continued fraction is denoted by a vinculum (a horizontal line above the block).

A reduced surd is a quadratic surd which is greater than 1 and whose conjugate is greater than −1 and less than 0. Galois proved that the continued fraction of a number is purely periodic, meaning it begins with its repeating block, if and only if that number is a reduced surd. He also proved that the repeating block of a reduced surd is the mirror image of the repeating block of the negative reciprocal of its conjugate, which must also be a reduced surd.[2]

A convergent is the truncation of a continued fraction. For example, the second convergent of {3; 2, 5, 6, 8} would be {3; 2}.

The Fibonacci numbers (Fn) are a sequence deﬁned by the recurrence Fn+2 = Fn+1 + Fn, where F0 = 0 and F1 = 1.

The Lucas numbers (Ln) are a sequence deﬁned by the recurrence Ln+2 = Ln+1 + Ln, where L0 = 2 and L1 = 1.

√

The golden ratio (φ) is 1+2 5 .

√

The silver means (Sm) are m+ 2m2+4 . the golden ratio, as S1 = φ.

The silver means are analogues to

The Lucas sequences are a family of sequences, consisting of two paired

types of sequences. The U-series (Un(P, Q) or simply Un) is deﬁned by the recurrence Un+2 = P Un+1 − QUn, where U0 = 0 and U1 = 1. The Vseries (Vn(P, Q) or Vn) is deﬁned by the recurrence Vn+2 = P Vn+1 − QVn, where V0 = 2 and V1 = P . The Lucas sequences are a generalization of the Fibonacci and Lucas nubmers, as Un(1, −1) = Fn and Vn(1, −1) = Ln.

2.2 New Deﬁnitions

The silver Fibonacci numbers (Fm,n) are a family of sequences deﬁned by the recurrence Fm,n+2 = mFm,n+1 + Fm,n, where Fm,0 = 0 and Fm,1 = 1. The

3

silver Fibonacci numbers are a generalization of the Fibonacci numbers, as F1,n = Fn.

The silver Lucas numbers (Lm,n) are a family of sequences deﬁned by the recurrence Lm,n+2 = mLm,n+1 + Lm,n, where Lm,0 = 2 and Lm,1 = m. The silver Lucas numbers are a generalization of the Lucas numbers, as L1,n = Ln.

The bronze Fibonacci numbers (fm,n) are a family of sequences deﬁned by the recurrence fm,n+2 = mfm,n+1 − fm,n, where fm,0 = 0 and fm,1 = 1.

The bronze Lucas numbers (lm,n) are a family of sequence deﬁned by the recurrence lm,n+2 = mlm,n+1 − lm,n, where lm,0 = 2 and lm,1 = m.

√

The bronze means (Bm) are m+ 2m2−4 . √

The Lucas constants (C(P, Q) or C) are deﬁned as P + P 2−4Q . The Lucas 2

constants are analogues to the golden ratio, as C(1, −1) = φ and generalizations of the silver and bronze means, as C(m, −1) = Sm and C(m, 1) = Bm. A Lucas constant is degenerate if P 2 − 4Q is a perfect square.

2.3 Known Properties

These are a few known properties of the Fibonacci and Lucas numbers and the golden ratio that will be used later on.

• Ln = Fn+1 + Fn−1 = 2Fn+1 − Fn. • Fn2 − Fn+1Fn−1 = (−1)n−1.

• The continued fraction for the golden ratio is {1; 1}.

• φn+2 = φn+1 + φn.

√

• φn = Ln+F2 n 5 .

• φφ = −1.

•

Fn =

φn−φn √

.

5

4

3 The Golden Ratio and Continued Fractions

The relationship between the golden ratio and continued fractions can be encompassed in three properties. The ﬁrst, which is commonly known, relates the convergents of the golden ratio to the Fibonacci numbers.[1] The second, which is known, but not as commonly, relates the powers of the golden ratio to the Lucas numbers.[3] The ﬁnal property pertains to the convergents of the powers of the golden ratio.

3.1 The Convergents of the Golden Ratio

Theorem 3.1. The nth convergent of the golden ratio is FFn+n1 .

Proof. We can easily prove this by induction. Clearly, this works for the case n = 1, as the 1st convergent is 1, and FF21 = 11 = 1. Assuming the nth convergent is FFn+n 1 , the n + 1th convergent is 1 + Fn1+1 = 1 + FFn+n 1 = FnF+n1++1Fn =

Fn

. Fn+2

Fn+1

3.2 The Powers of the Golden Ratio

Theorem 3.2. The continued fraction for φn is {Ln; Ln} if n is odd. The continued fraction for φn is {Ln − 1; 1, Ln − 2} if n is even.

Proof. Let x = {Ln; Ln} and assume n is odd.

x = Ln + x1 =⇒ x2 − Lnx − 1 = 0 =⇒ x = Ln + 2L2n + 4

= Ln +

(2Fn+1 − Fn)2 + 4 = Ln + 2

4Fn2+1 − 4Fn+1Fn + Fn2 + 4 2

= Ln + = Ln +

4Fn+1Fn−1 + Fn2 + 4 = Ln + 4(Fn+1Fn−1 − (−1)n) + Fn2

2

√

2

4Fn2 + Fn2 = Ln + Fn 5 = φn.

2

2

5

Now let x = {Ln − 1; 1, Ln − 2} and assume n is even.

x = Ln − 1 + 1 1 = Ln − 1 + x −x 1 = Ln − x1 =⇒ x2 − Lnx + 1 = 0 1+ x−1

=⇒ x = Ln + L2n − 4 = Ln + 2

(2Fn+1 − Fn)2 − 4 2

= Ln + = Ln + = φn.

4Fn2+1 − 4Fn+1Fn + Fn2 − 4 = Ln + 2

4(Fn+1Fn−1 − (−1)n) + Fn2 = Ln + 2

4Fn+1Fn−1 + Fn2 − 4

2

√

4Fn2 + Fn2 = Ln + Fn 5

2

2

3.3 The Convergents of the Powers of the Golden Ratio

Lemma 3.1. Fa(n+2) = LaFa(n+1) + Fan, when a is odd.

Proof. Let gn follow the proposed recurrence, that is let gn+2 = Lagn+1 + gn, ∞

and g0 = 0 and g1 = 1. Let G = gnxn. Then,

n=0

G = L xG + x2G + x =⇒ G(1 − L x − x2) = x =⇒ G =

x .

a

a

1 − Lax − x2

√ By solving the denominator, we ﬁnd its roots are −La± , L2a−4 which by sim-

2

ilar methods as the one used in the ﬁrst part of the previous proof, are equal to −φa and −φa.

x

− φa

φa

a

φa+x

− 1

1+

x a

1

1+

x a

− 1

1

1−φax

a

G=− a

= φ +x √

= φ √φ=

√1−φ x

(φ + x)(φa + x)

Fa 5

Fa 5

Fa 5

φan − φan Fan =⇒ gn = Fa√5 = Fa .

Since FFaan satisﬁes the recurrence, it follows that Fan would as well.

6

Lemma 3.2. Fa(n+2) = LaFa(n+1) − Fan, when a is even.

Proof. Let gn follow the proposed recurrence, that is let gn+2 = Lagn+1 − gn, ∞

and g0 = 0 and g1 = 1. Let G = gnxn. Then,

n=0

G = L xG − x2G + x =⇒ G(1 − L x + x2) = x =⇒ G =

x .

a

a

1 − Lax + x2

√

By solving the denominator, we ﬁnd its roots are La± L2a+4 , which are equal

to φa and φa.

2

x

− φa

φa

a

φa−x

− 1

1−

x a

1

1−

x a

− 1

1

1−φax

a

G= a

= φ −x √

= φ √φ=

√1−φ x

(φ − x)(φa − x)

Fa 5

Fa 5

Fa 5

φan − φan Fan =⇒ gn = Fa√5 = Fa .

Since FFaan satisﬁes the recurrence, it follows that Fan would as well.

Theorem 3.3. The nth convergent of φa is Fa(n+1) , if a is odd, and the 2nth Fan

convergent of φa is Fa(n+1) , if a is even. Fan

Proof. We can easily prove this by induction. For odd a, the case of n = 1

yields FF2aa = La, which is the ﬁrst convergent of φa = {La; La}. Assuming the nth convergent is FaF(na+n 1) , the n+1th convergent is La + Fa(n1+1) = La + FaF(na+n 1) =

Fan

LaFaF(an(+n+1)1+) Fan = FFaa((nn++21)) . For even a, the case of n = 1 yields FF2aa = La, which

is the second convergent of φa = {La − 1; 1, La − 2}. Assuming the 2nth

convergent is FaF(na+n1) , the 2n + 2th convergent is La − 1 + 1 + 1 1 = Fa(n+1) −1 Fan

1 Fa(n+1)−Fan

La Fa(n+1) −Fa(n+1) +Fa(n+1) −Fan

L −1+ = L −1+ = = a

Fan

a

Fa(n+1)

Fa(n+1)

1 + Fa(n+1)−Fan

= . LaFa(n+1)−Fan Fa(n+1)

Fa(n+2) Fa(n+1)

7

4 The Silver and Bronze Means

The properties we have found that relate the golden ratio and continued fractions can be generalized to a family of similar numbers, known as the silver means.[4] As the continued fraction of the golden ratio is {1; 1}, the continued fractions of the silver means are {m; m}. As the Fibonacci and Lucas numbers are related to the golden ratio, the families of sequences we will call the silver Fibonacci numbers and silver Lucas numbers are related to the golden ratio similarly. To complete the generalization, we must also deﬁne another family of constants and another two families of sequences, we will call the bronze means and the bronze Fibonacci and Lucas numbers. With these terms deﬁned, we can generalize theorems 3.1, 3.2, and 3.3.

4.1 Lemmas Pertaining to the Silver and Bronze Means

We will begin by stating lemmas analogous to the known properties listed in Section 2.3.

Lemma 4.1. Lm,n = Fm,n+1 + Fm,n−1 = 2Fm,n+1 − mFm,n. Similarly, lm,n = fm,n+1 − fm,n−1 = 2fm,n+1 − mfm,n. Proof. As both sides of each equation share the same recurrence, we need only show that the cases for n = 1 and n = 2 work.

Fm,2 + Fm,0 = m + 0 = Lm,1.

fm,2 − fm,0 = m − 0 = lm,1. Fm,3 + Fm,1 = m2 + 1 + 1 = m2 + 2 = Lm,2. fm,3 − fm,1 = m2 − 1 − 1 = m2 − 2 = lm,2.

Lemma 4.2. Fm2 ,n−Fm,n+1Fm,n−1 = (−1)n−1. Similarly, fm2 ,n−fm,n+1fm,n−1 = 1.

Proof. We can prove this by induction. This works for the case n = 1, as Fm2 ,1 − Fm,2Fm,0 = 1 − 0 = (−1)0 and fm2 ,1 − fm,2fm,0 = 1 − 0 = 1. Assuming

8

the property is true for n, we get

Fm2 ,n+1 − Fm,n+2Fm,n = Fm2 ,n+1 − Fm,n(mFm,n+1 + Fm,n) = Fm2 ,n+1 − mFm,n+1Fm,n − Fm2 ,n = Fm,n+1(Fm,n+1 − mFm,n) − Fm2 ,n = Fm,n+1Fm,n−1 − Fm2 ,n = −(−1)n−1 = (−1)n.

fm2 ,n+1 − fm,n+2fm,n = fm2 ,n+1 − fm,n(mfm,n+1 − fm,n) = fm2 ,n+1 − mfm,n+1fm,n + fm2 ,n = fm,n+1(fm,n+1 − mfm,n) + fm2 ,n = fm2 ,n − fm,n+1fm,n−1 = 1.

Lemma 4.3. The continued fractions for the silver means are {m; m}. The continued fractions for the bronze means are {m − 1; 1, m − 2}.

Proof. Let x = {m; m}. √

x = m + x1 =⇒ x2 − mx − 1 = 0 =⇒ x = m + 2m2 + 4 = Sm.

Now let x = {m − 1; 1, m − 2}.

x=m−1+

1

x−1

1

=m−1+

=m−

=⇒

x2 − mx + 1 = 0

1 +√x−1 1

x

x

m + m2 − 4

=⇒ x = 2 = Bm.

Lemma 4.4. Smn+2 = mSmn+1 + Smn . Similarly, Bmn+2 = mBmn+1 − Bmn .

Proof. It is clear from the deﬁnition of the silver and bronze means that Sm2 = mSm + 1 and Bm2 = mBm − 1, so by multiplying the equations by Smn and Bmn respectively, it follows quite simply that Smn+2 = mSmn+1 + Smn and Bmn+2 = mBmn+1 − Bmn .

9

√

√

Lemma

4.5.

Smn

=

Lm,n +Fm,n 2

m2+4 .

Similarly,

Bmn

=

lm,n +fm,n 2

m2−4 .

Proof. As both sides of each equation share the same recurrence, we need

only show that the cases for n = 0 and n = 1 work.

√

√

Lm,0 + Fm,0 m2 + 4 = 2 + 0 · m2 + 4 = 1 = S0 .

2

2

m

√

√

Lm,1 + Fm,1 m2 + 4 = m + 1 · m2 + 4 = S1 .

2

2

m

√

√

lm,0 + fm,0 m2 − 4 = 2 + 0 · m2 − 4 = 1 = B0 .

2

2

m

√

√

lm,1 + fm,1 m2 − 4 = m + 1 · m2 − 4 = B1 .

2

2

m

Lemma 4.6. SmSm = −1. Similarly, BmBm = 1.

Proof.

√

√

m + m2 + 4 m − m2 + 4 m2 − m2 − 4

SmSm = 2 · 2 = 4 = −1.

√

√

m + m2 − 4 m − m2 − 4 m2 − m2 + 4

BmBm = 2 · 2 = 4 = 1.

Lemma 4.7. Fm,n = S√m nm−2S+m4n . Similarly, fm,n = B√m nm−2B−m4n .

Proof.

Sn − S n

− √

Lm,n+Fm,n m2+4

√ Lm,n−Fm,n m2+4

√ F m2 + 4

√m m =

2√

2

= m√,n

= Fm,n.

m2 + 4

m2 + 4

m2 + 4

Bn − B n

− √

lm,n+fm,n m2−4

√ lm,n−fm,n m2−4

√ f m2 − 4

√m m =

2√

2

= m√,n

= fm,n.

m2 − 4

m2 − 4

m2 − 4

10

Convergents

Saarik Kalia

December 23, 2011

Abstract The relationship between the golden ratio and continued fractions is commonly known about throughout the mathematical world: the convergents of the continued fraction are the ratios of consecutive Fibonacci numbers. The continued fractions for the powers of the golden ratio also exhibit an interesting relationship with the Lucas numbers. In this paper, we study the silver means and introduce the bronze means, which are generalizations of the golden ratio. We correspondingly introduce the silver and bronze Fibonacci and Lucas numbers, and we prove the relationship between the convergents of the continued fractions of the powers of the silver and bronze means and the silver and bronze Fibonacci and Lucas numbers. We further generalize this to the Lucas constants, a two-parameter generalization of the golden ratio.

1 Introduction

The golden ratio exhibits an interesting relationship with continued fractions. We can summarize this relationship in three already known properties. Firstly, the nth convergent of the golden ratio is Fn+1 .[1] Secondly, the con-

Fn

tinued fraction of the nth power of the golden ratio is {Ln; Ln} for odd n and {Ln − 1; 1, Ln − 2} for even n.[3] Finally, the convergents of the powers of the

1

golden ratio can be expressed as Fa(n+1) . In this paper, we will generalize the Fan

golden ratio to a group of constants and see that corresponding properties hold.

In Section 2, we will deﬁne old and new terms and list known properties. In Section 3, we will prove the three properties, which constitute the relationship between the golden ratio and continued fractions. In Section 4, we generalize this to the silver means, whose continued fractions are {m; m}, analogous to the golden ratio’s {1; 1}. To complete the generalization, not only must we generalize the Fibonacci and Lucas numbers to two families of series we will call the silver Fibonacci and Lucas numbers, but we need to also deﬁne the bronze means and the associated bronze Fibonacci and Lucas numbers. We will ﬁnd that the three properties which were true of the golden ratio also hold for the silver and bronze means. In Section 5, we will go even further to a two-parameter generalization of these properties. The Lucas sequences are a family of sequences, of which the silver and bronze Fibonacci and Lucas numbers are a subset. When we deﬁne the Lucas constants, constants associated with a Lucas sequence and analogous to the silver and bronze means, we see that they obey one of the properties. The other two, however, only certain Lucas constants obey. We will discuss the nature of the continued fractions of the Lucas constants and for which Lucas constants these properties hold. Though we will ﬁnd certain cases for which these properties hold, the search for an all-encompassing case will be left to further research.

2 Deﬁnitions and Properties

2.1 Old Deﬁnitions

A continued fraction is a form of representing a number by nested fractions,

all of whose numerators are 1. For instance, the continued fraction for 79

1

is 1 +

. The compact notation for this continued fraction is {1; 3, 2}.

1

3+

2

(Note a semicolon follows the ﬁrst term, while commas follow the others.)

The continued fraction of a number is ﬁnite if and only if that number is

rational.

2

A quadratic irrational or quadratic surd is a number that is the solution to some quadratic equation with rational coeﬃcients. The continued fraction of a number is periodic, meaning it has a repeating block, if and only if that number is a quadratic irrational. The repeating block of a periodic continued fraction is denoted by a vinculum (a horizontal line above the block).

A reduced surd is a quadratic surd which is greater than 1 and whose conjugate is greater than −1 and less than 0. Galois proved that the continued fraction of a number is purely periodic, meaning it begins with its repeating block, if and only if that number is a reduced surd. He also proved that the repeating block of a reduced surd is the mirror image of the repeating block of the negative reciprocal of its conjugate, which must also be a reduced surd.[2]

A convergent is the truncation of a continued fraction. For example, the second convergent of {3; 2, 5, 6, 8} would be {3; 2}.

The Fibonacci numbers (Fn) are a sequence deﬁned by the recurrence Fn+2 = Fn+1 + Fn, where F0 = 0 and F1 = 1.

The Lucas numbers (Ln) are a sequence deﬁned by the recurrence Ln+2 = Ln+1 + Ln, where L0 = 2 and L1 = 1.

√

The golden ratio (φ) is 1+2 5 .

√

The silver means (Sm) are m+ 2m2+4 . the golden ratio, as S1 = φ.

The silver means are analogues to

The Lucas sequences are a family of sequences, consisting of two paired

types of sequences. The U-series (Un(P, Q) or simply Un) is deﬁned by the recurrence Un+2 = P Un+1 − QUn, where U0 = 0 and U1 = 1. The Vseries (Vn(P, Q) or Vn) is deﬁned by the recurrence Vn+2 = P Vn+1 − QVn, where V0 = 2 and V1 = P . The Lucas sequences are a generalization of the Fibonacci and Lucas nubmers, as Un(1, −1) = Fn and Vn(1, −1) = Ln.

2.2 New Deﬁnitions

The silver Fibonacci numbers (Fm,n) are a family of sequences deﬁned by the recurrence Fm,n+2 = mFm,n+1 + Fm,n, where Fm,0 = 0 and Fm,1 = 1. The

3

silver Fibonacci numbers are a generalization of the Fibonacci numbers, as F1,n = Fn.

The silver Lucas numbers (Lm,n) are a family of sequences deﬁned by the recurrence Lm,n+2 = mLm,n+1 + Lm,n, where Lm,0 = 2 and Lm,1 = m. The silver Lucas numbers are a generalization of the Lucas numbers, as L1,n = Ln.

The bronze Fibonacci numbers (fm,n) are a family of sequences deﬁned by the recurrence fm,n+2 = mfm,n+1 − fm,n, where fm,0 = 0 and fm,1 = 1.

The bronze Lucas numbers (lm,n) are a family of sequence deﬁned by the recurrence lm,n+2 = mlm,n+1 − lm,n, where lm,0 = 2 and lm,1 = m.

√

The bronze means (Bm) are m+ 2m2−4 . √

The Lucas constants (C(P, Q) or C) are deﬁned as P + P 2−4Q . The Lucas 2

constants are analogues to the golden ratio, as C(1, −1) = φ and generalizations of the silver and bronze means, as C(m, −1) = Sm and C(m, 1) = Bm. A Lucas constant is degenerate if P 2 − 4Q is a perfect square.

2.3 Known Properties

These are a few known properties of the Fibonacci and Lucas numbers and the golden ratio that will be used later on.

• Ln = Fn+1 + Fn−1 = 2Fn+1 − Fn. • Fn2 − Fn+1Fn−1 = (−1)n−1.

• The continued fraction for the golden ratio is {1; 1}.

• φn+2 = φn+1 + φn.

√

• φn = Ln+F2 n 5 .

• φφ = −1.

•

Fn =

φn−φn √

.

5

4

3 The Golden Ratio and Continued Fractions

The relationship between the golden ratio and continued fractions can be encompassed in three properties. The ﬁrst, which is commonly known, relates the convergents of the golden ratio to the Fibonacci numbers.[1] The second, which is known, but not as commonly, relates the powers of the golden ratio to the Lucas numbers.[3] The ﬁnal property pertains to the convergents of the powers of the golden ratio.

3.1 The Convergents of the Golden Ratio

Theorem 3.1. The nth convergent of the golden ratio is FFn+n1 .

Proof. We can easily prove this by induction. Clearly, this works for the case n = 1, as the 1st convergent is 1, and FF21 = 11 = 1. Assuming the nth convergent is FFn+n 1 , the n + 1th convergent is 1 + Fn1+1 = 1 + FFn+n 1 = FnF+n1++1Fn =

Fn

. Fn+2

Fn+1

3.2 The Powers of the Golden Ratio

Theorem 3.2. The continued fraction for φn is {Ln; Ln} if n is odd. The continued fraction for φn is {Ln − 1; 1, Ln − 2} if n is even.

Proof. Let x = {Ln; Ln} and assume n is odd.

x = Ln + x1 =⇒ x2 − Lnx − 1 = 0 =⇒ x = Ln + 2L2n + 4

= Ln +

(2Fn+1 − Fn)2 + 4 = Ln + 2

4Fn2+1 − 4Fn+1Fn + Fn2 + 4 2

= Ln + = Ln +

4Fn+1Fn−1 + Fn2 + 4 = Ln + 4(Fn+1Fn−1 − (−1)n) + Fn2

2

√

2

4Fn2 + Fn2 = Ln + Fn 5 = φn.

2

2

5

Now let x = {Ln − 1; 1, Ln − 2} and assume n is even.

x = Ln − 1 + 1 1 = Ln − 1 + x −x 1 = Ln − x1 =⇒ x2 − Lnx + 1 = 0 1+ x−1

=⇒ x = Ln + L2n − 4 = Ln + 2

(2Fn+1 − Fn)2 − 4 2

= Ln + = Ln + = φn.

4Fn2+1 − 4Fn+1Fn + Fn2 − 4 = Ln + 2

4(Fn+1Fn−1 − (−1)n) + Fn2 = Ln + 2

4Fn+1Fn−1 + Fn2 − 4

2

√

4Fn2 + Fn2 = Ln + Fn 5

2

2

3.3 The Convergents of the Powers of the Golden Ratio

Lemma 3.1. Fa(n+2) = LaFa(n+1) + Fan, when a is odd.

Proof. Let gn follow the proposed recurrence, that is let gn+2 = Lagn+1 + gn, ∞

and g0 = 0 and g1 = 1. Let G = gnxn. Then,

n=0

G = L xG + x2G + x =⇒ G(1 − L x − x2) = x =⇒ G =

x .

a

a

1 − Lax − x2

√ By solving the denominator, we ﬁnd its roots are −La± , L2a−4 which by sim-

2

ilar methods as the one used in the ﬁrst part of the previous proof, are equal to −φa and −φa.

x

− φa

φa

a

φa+x

− 1

1+

x a

1

1+

x a

− 1

1

1−φax

a

G=− a

= φ +x √

= φ √φ=

√1−φ x

(φ + x)(φa + x)

Fa 5

Fa 5

Fa 5

φan − φan Fan =⇒ gn = Fa√5 = Fa .

Since FFaan satisﬁes the recurrence, it follows that Fan would as well.

6

Lemma 3.2. Fa(n+2) = LaFa(n+1) − Fan, when a is even.

Proof. Let gn follow the proposed recurrence, that is let gn+2 = Lagn+1 − gn, ∞

and g0 = 0 and g1 = 1. Let G = gnxn. Then,

n=0

G = L xG − x2G + x =⇒ G(1 − L x + x2) = x =⇒ G =

x .

a

a

1 − Lax + x2

√

By solving the denominator, we ﬁnd its roots are La± L2a+4 , which are equal

to φa and φa.

2

x

− φa

φa

a

φa−x

− 1

1−

x a

1

1−

x a

− 1

1

1−φax

a

G= a

= φ −x √

= φ √φ=

√1−φ x

(φ − x)(φa − x)

Fa 5

Fa 5

Fa 5

φan − φan Fan =⇒ gn = Fa√5 = Fa .

Since FFaan satisﬁes the recurrence, it follows that Fan would as well.

Theorem 3.3. The nth convergent of φa is Fa(n+1) , if a is odd, and the 2nth Fan

convergent of φa is Fa(n+1) , if a is even. Fan

Proof. We can easily prove this by induction. For odd a, the case of n = 1

yields FF2aa = La, which is the ﬁrst convergent of φa = {La; La}. Assuming the nth convergent is FaF(na+n 1) , the n+1th convergent is La + Fa(n1+1) = La + FaF(na+n 1) =

Fan

LaFaF(an(+n+1)1+) Fan = FFaa((nn++21)) . For even a, the case of n = 1 yields FF2aa = La, which

is the second convergent of φa = {La − 1; 1, La − 2}. Assuming the 2nth

convergent is FaF(na+n1) , the 2n + 2th convergent is La − 1 + 1 + 1 1 = Fa(n+1) −1 Fan

1 Fa(n+1)−Fan

La Fa(n+1) −Fa(n+1) +Fa(n+1) −Fan

L −1+ = L −1+ = = a

Fan

a

Fa(n+1)

Fa(n+1)

1 + Fa(n+1)−Fan

= . LaFa(n+1)−Fan Fa(n+1)

Fa(n+2) Fa(n+1)

7

4 The Silver and Bronze Means

The properties we have found that relate the golden ratio and continued fractions can be generalized to a family of similar numbers, known as the silver means.[4] As the continued fraction of the golden ratio is {1; 1}, the continued fractions of the silver means are {m; m}. As the Fibonacci and Lucas numbers are related to the golden ratio, the families of sequences we will call the silver Fibonacci numbers and silver Lucas numbers are related to the golden ratio similarly. To complete the generalization, we must also deﬁne another family of constants and another two families of sequences, we will call the bronze means and the bronze Fibonacci and Lucas numbers. With these terms deﬁned, we can generalize theorems 3.1, 3.2, and 3.3.

4.1 Lemmas Pertaining to the Silver and Bronze Means

We will begin by stating lemmas analogous to the known properties listed in Section 2.3.

Lemma 4.1. Lm,n = Fm,n+1 + Fm,n−1 = 2Fm,n+1 − mFm,n. Similarly, lm,n = fm,n+1 − fm,n−1 = 2fm,n+1 − mfm,n. Proof. As both sides of each equation share the same recurrence, we need only show that the cases for n = 1 and n = 2 work.

Fm,2 + Fm,0 = m + 0 = Lm,1.

fm,2 − fm,0 = m − 0 = lm,1. Fm,3 + Fm,1 = m2 + 1 + 1 = m2 + 2 = Lm,2. fm,3 − fm,1 = m2 − 1 − 1 = m2 − 2 = lm,2.

Lemma 4.2. Fm2 ,n−Fm,n+1Fm,n−1 = (−1)n−1. Similarly, fm2 ,n−fm,n+1fm,n−1 = 1.

Proof. We can prove this by induction. This works for the case n = 1, as Fm2 ,1 − Fm,2Fm,0 = 1 − 0 = (−1)0 and fm2 ,1 − fm,2fm,0 = 1 − 0 = 1. Assuming

8

the property is true for n, we get

Fm2 ,n+1 − Fm,n+2Fm,n = Fm2 ,n+1 − Fm,n(mFm,n+1 + Fm,n) = Fm2 ,n+1 − mFm,n+1Fm,n − Fm2 ,n = Fm,n+1(Fm,n+1 − mFm,n) − Fm2 ,n = Fm,n+1Fm,n−1 − Fm2 ,n = −(−1)n−1 = (−1)n.

fm2 ,n+1 − fm,n+2fm,n = fm2 ,n+1 − fm,n(mfm,n+1 − fm,n) = fm2 ,n+1 − mfm,n+1fm,n + fm2 ,n = fm,n+1(fm,n+1 − mfm,n) + fm2 ,n = fm2 ,n − fm,n+1fm,n−1 = 1.

Lemma 4.3. The continued fractions for the silver means are {m; m}. The continued fractions for the bronze means are {m − 1; 1, m − 2}.

Proof. Let x = {m; m}. √

x = m + x1 =⇒ x2 − mx − 1 = 0 =⇒ x = m + 2m2 + 4 = Sm.

Now let x = {m − 1; 1, m − 2}.

x=m−1+

1

x−1

1

=m−1+

=m−

=⇒

x2 − mx + 1 = 0

1 +√x−1 1

x

x

m + m2 − 4

=⇒ x = 2 = Bm.

Lemma 4.4. Smn+2 = mSmn+1 + Smn . Similarly, Bmn+2 = mBmn+1 − Bmn .

Proof. It is clear from the deﬁnition of the silver and bronze means that Sm2 = mSm + 1 and Bm2 = mBm − 1, so by multiplying the equations by Smn and Bmn respectively, it follows quite simply that Smn+2 = mSmn+1 + Smn and Bmn+2 = mBmn+1 − Bmn .

9

√

√

Lemma

4.5.

Smn

=

Lm,n +Fm,n 2

m2+4 .

Similarly,

Bmn

=

lm,n +fm,n 2

m2−4 .

Proof. As both sides of each equation share the same recurrence, we need

only show that the cases for n = 0 and n = 1 work.

√

√

Lm,0 + Fm,0 m2 + 4 = 2 + 0 · m2 + 4 = 1 = S0 .

2

2

m

√

√

Lm,1 + Fm,1 m2 + 4 = m + 1 · m2 + 4 = S1 .

2

2

m

√

√

lm,0 + fm,0 m2 − 4 = 2 + 0 · m2 − 4 = 1 = B0 .

2

2

m

√

√

lm,1 + fm,1 m2 − 4 = m + 1 · m2 − 4 = B1 .

2

2

m

Lemma 4.6. SmSm = −1. Similarly, BmBm = 1.

Proof.

√

√

m + m2 + 4 m − m2 + 4 m2 − m2 − 4

SmSm = 2 · 2 = 4 = −1.

√

√

m + m2 − 4 m − m2 − 4 m2 − m2 + 4

BmBm = 2 · 2 = 4 = 1.

Lemma 4.7. Fm,n = S√m nm−2S+m4n . Similarly, fm,n = B√m nm−2B−m4n .

Proof.

Sn − S n

− √

Lm,n+Fm,n m2+4

√ Lm,n−Fm,n m2+4

√ F m2 + 4

√m m =

2√

2

= m√,n

= Fm,n.

m2 + 4

m2 + 4

m2 + 4

Bn − B n

− √

lm,n+fm,n m2−4

√ lm,n−fm,n m2−4

√ f m2 − 4

√m m =

2√

2

= m√,n

= fm,n.

m2 − 4

m2 − 4

m2 − 4

10

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