Supersilver ratio

Algebraic integer, approximately 2.20557
Supersilver ratio
A supersilver rectangle contains two scaled copies of itself, ς = ((ς − 1)2 + 2(ς − 1) + 1) / ς
Rationalityirrational algebraic
Symbolς
Representations
Decimal2.2055694304005903117020286...
Algebraic formreal root of x3 = 2x2 + 1
Continued fraction (linear)[2;4,1,6,2,1,1,1,1,1,1,2,2,1,2,1,...]
not periodic
infinite

In mathematics, the supersilver ratio is a geometrical proportion close to 75/34. Its true value is the real solution of the equation x3 = 2x2 + 1.

The name supersilver ratio results from analogy with the silver ratio, the positive solution of the equation x2 = 2x + 1, and the supergolden ratio.

Definition

Two quantities a > b > 0 are in the supersilver ratio-squared if

( 2 a + b a ) 2 = a b {\displaystyle \left({\frac {2a+b}{a}}\right)^{2}={\frac {a}{b}}} .

The ratio 2 a + b a {\displaystyle {\frac {2a+b}{a}}} is here denoted ς . {\displaystyle \varsigma .}

Based on this definition, one has

1 = ( 2 a + b a ) 2 b a = ( 2 a + b a ) 2 ( 2 a + b a − 2 ) ⟹ ς 2 ( ς − 2 ) = 1 {\displaystyle {\begin{aligned}1&=\left({\frac {2a+b}{a}}\right)^{2}{\frac {b}{a}}\\&=\left({\frac {2a+b}{a}}\right)^{2}\left({\frac {2a+b}{a}}-2\right)\\&\implies \varsigma ^{2}\left(\varsigma -2\right)=1\end{aligned}}}

It follows that the supersilver ratio is found as the unique real solution of the cubic equation ς 3 − 2 ς 2 − 1 = 0. {\displaystyle \varsigma ^{3}-2\varsigma ^{2}-1=0.} The decimal expansion of the root begins as 2.205 569 430 400 590... {\displaystyle 2.205\,569\,430\,400\,590...} (sequence A356035 in the OEIS).

The minimal polynomial for the reciprocal root is the depressed cubic x 3 + 2 x − 1 , {\displaystyle x^{3}+2x-1,} thus the simplest solution with Cardano's formula,

w 1 , 2 = ( 1 ± 1 3 59 3 ) / 2 {\displaystyle w_{1,2}=\left(1\pm {\frac {1}{3}}{\sqrt {\frac {59}{3}}}\right)/2}
1 / ς = w 1 3 + w 2 3 {\displaystyle 1/\varsigma ={\sqrt[{3}]{w_{1}}}+{\sqrt[{3}]{w_{2}}}}

or, using the hyperbolic sine,

1 / ς = − 2 2 3 sinh ⁡ ( 1 3 arsinh ⁡ ( − 3 4 3 2 ) ) . {\displaystyle 1/\varsigma =-2{\sqrt {\frac {2}{3}}}\sinh \left({\frac {1}{3}}\operatorname {arsinh} \left(-{\frac {3}{4}}{\sqrt {\frac {3}{2}}}\right)\right).}

1 / ς {\displaystyle 1/\varsigma } is the superstable fixed point of the iteration x ← ( 2 x 3 + 1 ) / ( 3 x 2 + 2 ) . {\displaystyle x\gets (2x^{3}+1)/(3x^{2}+2).}

Rewrite the minimal polynomial as ( x 2 + 1 ) 2 = 1 + x {\displaystyle (x^{2}+1)^{2}=1+x} , then the iteration x ← − 1 + 1 + x {\displaystyle x\gets {\sqrt {-1+{\sqrt {1+x}}}}} results in the continued radical

1 / ς = − 1 + 1 + − 1 + 1 + ⋯ {\displaystyle 1/\varsigma ={\sqrt {-1+{\sqrt {1+{\sqrt {-1+{\sqrt {1+\cdots }}}}}}}}\;} [1]

Dividing the defining trinomial x 3 − 2 x 2 − 1 {\displaystyle x^{3}-2x^{2}-1} by x − ς {\displaystyle x-\varsigma } one obtains x 2 + x / ς 2 + 1 / ς {\displaystyle x^{2}+x/\varsigma ^{2}+1/\varsigma } , and the conjugate elements of ς {\displaystyle \varsigma } are

x 1 , 2 = ( − 1 ± i 8 ς 2 + 3 ) / 2 ς 2 {\displaystyle x_{1,2}=\left(-1\pm i{\sqrt {8\varsigma ^{2}+3}}\right)/2\varsigma ^{2}}

Properties

Rectangles with aspect ratios related to powers of ς tile the square.

The growth rate of the average value of the n-th term of a random Fibonacci sequence is ς − 1 {\displaystyle \varsigma -1} .[2]

The supersilver ratio can be expressed in terms of itself as the infinite geometric series

ς = 2 ∑ k = 0 ∞ ς − 3 k {\displaystyle \varsigma =2\sum _{k=0}^{\infty }\varsigma ^{-3k}} and ς 2 = − 1 + ∑ k = 0 ∞ ( ς − 1 ) − k , {\displaystyle \,\varsigma ^{2}=-1+\sum _{k=0}^{\infty }(\varsigma -1)^{-k},}

in comparison to the silver ratio identities

σ = 2 ∑ k = 0 ∞ σ − 2 k {\displaystyle \sigma =2\sum _{k=0}^{\infty }\sigma ^{-2k}} and σ 2 = − 1 + 2 ∑ k = 0 ∞ ( σ − 1 ) − k . {\displaystyle \,\sigma ^{2}=-1+2\sum _{k=0}^{\infty }(\sigma -1)^{-k}.}

For every integer n {\displaystyle n} one has

ς n = 2 ς n − 1 + ς n − 3 = 4 ς n − 2 + ς n − 3 + 2 ς n − 4 = ς n − 1 + 2 ς n − 2 + ς n − 3 + ς n − 4 . {\displaystyle {\begin{aligned}\varsigma ^{n}&=2\varsigma ^{n-1}+\varsigma ^{n-3}\\&=4\varsigma ^{n-2}+\varsigma ^{n-3}+2\varsigma ^{n-4}\\&=\varsigma ^{n-1}+2\varsigma ^{n-2}+\varsigma ^{n-3}+\varsigma ^{n-4}.\end{aligned}}}

Continued fraction pattern of a few low powers

ς − 2 = [ 0 ; 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , . . . ] ≈ 0.2056 {\displaystyle \varsigma ^{-2}=[0;4,1,6,2,1,1,1,1,1,1,...]\approx 0.2056} (5/24)
ς − 1 = [ 0 ; 2 , 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , . . . ] ≈ 0.4534 {\displaystyle \varsigma ^{-1}=[0;2,4,1,6,2,1,1,1,1,1,...]\approx 0.4534} (5/11)
  ς 0 = [ 1 ] {\displaystyle \ \varsigma ^{0}=[1]}
ς 1 = [ 2 ; 4 , 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , . . . ] ≈ 2.2056 {\displaystyle \varsigma ^{1}=[2;4,1,6,2,1,1,1,1,1,1,...]\approx 2.2056} (53/24)
ς 2 = [ 4 ; 1 , 6 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , . . . ] ≈ 4.8645 {\displaystyle \varsigma ^{2}=[4;1,6,2,1,1,1,1,1,1,2,...]\approx 4.8645} (73/15)
ς 3 = [ 10 ; 1 , 2 , 1 , 2 , 4 , 4 , 2 , 2 , 6 , 2 , . . . ] ≈ 10.729 {\displaystyle \varsigma ^{3}=[10;1,2,1,2,4,4,2,2,6,2,...]\approx 10.729} (118/11)

The supersilver ratio is a Pisot number.[3] Because the absolute value 1 / ς {\displaystyle 1/{\sqrt {\varsigma }}} of the algebraic conjugates is smaller than 1, powers of ς {\displaystyle \varsigma } generate almost integers. For example: ς 10 = 2724.00146856... ≈ 2724 + 1 / 681. {\displaystyle \varsigma ^{10}=2724.00146856...\approx 2724+1/681.} After ten rotation steps the phases of the inward spiraling conjugate pair – initially close to ± 45 π / 82 {\displaystyle \pm 45\pi /82} – nearly align with the imaginary axis.

The minimal polynomial of the supersilver ratio m ( x ) = x 3 − 2 x 2 − 1 {\displaystyle m(x)=x^{3}-2x^{2}-1} has discriminant Δ = − 59 {\displaystyle \Delta =-59} and factors into ( x − 19 ) ( x − 21 ) 2 ( mod 59 ) . {\displaystyle (x-19)(x-21)^{2}{\pmod {59}}.} The imaginary quadratic field K = Q ( Δ ) {\displaystyle K=\mathbb {Q} ({\sqrt {\Delta }})} has class number h = 3. {\displaystyle h=3.} Then the Hilbert class field of K {\displaystyle K} can be formed by adjoining ς {\displaystyle \varsigma } .[4] With argument τ = ( 1 + Δ ) / 2 {\displaystyle \tau =(1+{\sqrt {\Delta }})/2\,} a generator for the ring of integers of K {\displaystyle K} , the real root  j(τ) of the Hilbert class polynomial is given by ( ς − 6 − 27 ς 6 − 6 ) 3 . {\displaystyle (\varsigma ^{-6}-27\varsigma ^{6}-6)^{3}.} [5][6]

The Weber-Ramanujan class invariant is approximated with error < 3.5 ∙ 10−20 by

2 f ( Δ ) = 2 4 G 59 ≈ ( e π − Δ + 24 ) 1 / 24 , {\displaystyle {\sqrt {2}}\,{\mathfrak {f}}({\sqrt {\Delta }})={\sqrt[{4}]{2}}\,G_{59}\approx (e^{\pi {\sqrt {-\Delta }}}+24)^{1/24},}

while its true value is the single real root of the polynomial

W 59 ( x ) = x 9 − 4 x 8 + 4 x 7 − 2 x 6 + 4 x 5 − 8 x 4 + 4 x 3 − 8 x 2 + 16 x − 8. {\displaystyle W_{59}(x)=x^{9}-4x^{8}+4x^{7}-2x^{6}+4x^{5}-8x^{4}+4x^{3}-8x^{2}+16x-8.}

The elliptic integral singular value[7] k r = λ ∗ ( r ) {\displaystyle k_{r}=\lambda ^{*}(r)} for r = 59 {\displaystyle r=59} has closed form expression

λ ∗ ( 59 ) = sin ⁡ ( arcsin ⁡ ( G 59 − 12 ) / 2 ) {\displaystyle \lambda ^{*}(59)=\sin(\arcsin \left(G_{59}^{-12}\right)/2)}

(which is less than 1/294 the eccentricity of the orbit of Venus).

Third-order Pell sequences

These numbers are associated with the supersilver ratio as the Pell numbers and Pell-Lucas numbers are with the silver ratio.

The fundamental sequence is defined by the third-order recurrence relation

S n = 2 S n − 1 + S n − 3 {\displaystyle S_{n}=2S_{n-1}+S_{n-3}} for n > 2,

with initial values

S 0 = 1 , S 1 = 2 , S 2 = 4. {\displaystyle S_{0}=1,S_{1}=2,S_{2}=4.}

The first few terms are 1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064,... (sequence A008998 in the OEIS). The limit ratio between consecutive terms is the supersilver ratio.

The first 8 indices n for which S n {\displaystyle S_{n}} is prime are n = 1, 6, 21, 114, 117, 849, 2418, 6144. The last number has 2111 decimal digits.

The sequence can be extended to negative indices using

S n = S n + 3 − 2 S n + 2 . {\displaystyle S_{n}=S_{n+3}-2S_{n+2}.}

The generating function of the sequence is given by

1 1 − 2 x − x 3 = ∑ n = 0 ∞ S n x n {\displaystyle {\frac {1}{1-2x-x^{3}}}=\sum _{n=0}^{\infty }S_{n}x^{n}} for x < 1 / ς {\displaystyle x<1/\varsigma \;} .[8]

The third-order Pell numbers are related to sums of binomial coefficients by

S n = ∑ k = 0 ⌊ n / 3 ⌋ ( n − 2 k k ) ⋅ 2 n − 3 k {\displaystyle S_{n}=\sum _{k=0}^{\lfloor n/3\rfloor }{n-2k \choose k}\cdot 2^{n-3k}\;} .[9]

The characteristic equation of the recurrence is x 3 − 2 x 2 − 1 = 0. {\displaystyle x^{3}-2x^{2}-1=0.} If the three solutions are real root α {\displaystyle \alpha } and conjugate pair β {\displaystyle \beta } and γ {\displaystyle \gamma } , the supersilver numbers can be computed with the Binet formula

S n − 2 = a α n + b β n + c γ n , {\displaystyle S_{n-2}=a\alpha ^{n}+b\beta ^{n}+c\gamma ^{n},} with real a {\displaystyle a} and conjugates b {\displaystyle b} and c {\displaystyle c} the roots of 59 x 3 + 4 x − 1 = 0. {\displaystyle 59x^{3}+4x-1=0.}

Since | b β n + c γ n | < 1 / α n {\displaystyle \left\vert b\beta ^{n}+c\gamma ^{n}\right\vert <1/{\sqrt {\alpha ^{n}}}} and α = ς , {\displaystyle \alpha =\varsigma ,} the number S n {\displaystyle S_{n}} is the nearest integer to a ς n + 2 , {\displaystyle a\,\varsigma ^{n+2},} with n ≥ 0 and a = ς / ( 2 ς 2 + 3 ) = {\displaystyle a=\varsigma /(2\varsigma ^{2}+3)=} 0.1732702315504081807484794...

Coefficients a = b = c = 1 {\displaystyle a=b=c=1} result in the Binet formula for the related sequence A n = S n + 2 S n − 3 . {\displaystyle A_{n}=S_{n}+2S_{n-3}.}

The first few terms are 3, 2, 4, 11, 24, 52, 115, 254, 560, 1235, 2724, 6008,... (sequence A332647 in the OEIS).

This third-order Pell-Lucas sequence has the Fermat property: if p is prime, A p ≡ A 1 mod p . {\displaystyle A_{p}\equiv A_{1}{\bmod {p}}.} The converse does not hold, but the small number of odd pseudoprimes n ∣ ( A n − 2 ) {\displaystyle \,n\mid (A_{n}-2)} makes the sequence special.[10] The 14 odd composite numbers below 108 to pass the test are n = 32, 52, 53, 315, 99297, 222443, 418625, 9122185, 32572, 11889745, 20909625, 24299681, 64036831, 76917325.

The third-order Pell numbers are obtained as integral powers n > 3 of a matrix with real eigenvalue ς {\displaystyle \varsigma }

Q = ( 2 0 1 1 0 0 0 1 0 ) , {\displaystyle Q={\begin{pmatrix}2&0&1\\1&0&0\\0&1&0\end{pmatrix}},}
Q n = ( S n S n − 2 S n − 1 S n − 1 S n − 3 S n − 2 S n − 2 S n − 4 S n − 3 ) {\displaystyle Q^{n}={\begin{pmatrix}S_{n}&S_{n-2}&S_{n-1}\\S_{n-1}&S_{n-3}&S_{n-2}\\S_{n-2}&S_{n-4}&S_{n-3}\end{pmatrix}}}

The trace of Q n {\displaystyle Q^{n}} gives the above A n . {\displaystyle A_{n}.}

Supersilver rectangle

Powers of ς within a supersilver rectangle.

A supersilver rectangle is a rectangle whose side lengths are in a ς : 1 {\displaystyle \varsigma :1} ratio. Compared to the silver rectangle, containing a single scaled copy of itself, the supersilver rectangle has one more degree of self-similarity.

Given a rectangle of height 1 and length ς {\displaystyle \varsigma } . On the right-hand side, cut off a square of side length 1 and mark the intersection with the falling diagonal. The remaining rectangle now has aspect ratio 1 + 1 / ς 2 : 1 {\displaystyle 1+1/\varsigma ^{2}:1} (according to ς = 2 + 1 / ς 2 {\displaystyle \varsigma =2+1/\varsigma ^{2}} ). Divide the original rectangle into four parts by a second, horizontal cut passing through the intersection point.[11]

Along the diagonal are two supersilver rectangles. The original rectangle and the scaled copies have diagonal lengths in the ratios ς / ( ς − 1 ) : ( ς − 1 ) : 1 {\displaystyle \varsigma /(\varsigma -1):(\varsigma -1):1} or, equivalently, areas ς 2 / ( ς − 1 ) 2 : ( ς − 1 ) 2 : 1. {\displaystyle \varsigma ^{2}/(\varsigma -1)^{2}:(\varsigma -1)^{2}:1.} The areas of the rectangles opposite the diagonal are both equal to ( ς − 1 ) / ς , {\displaystyle (\varsigma -1)/\varsigma ,} with aspect ratios ς + 1 / ς {\displaystyle \varsigma +1/\varsigma } (below) and ς / ( ς − 1 ) {\displaystyle \varsigma /(\varsigma -1)} (above).

The process can be repeated in the smallest supersilver rectangle at a scale of 1 : ς . {\displaystyle 1:\varsigma .}

See also

  • Solutions of equations similar to x 3 = 2 x 2 + 1 {\displaystyle x^{3}=2x^{2}+1} :
    • Silver ratio – the only positive solution of the equation x 2 = 2 x + 1 {\displaystyle x^{2}=2x+1}
    • Golden ratio – the only positive solution of the equation x 2 = x + 1 {\displaystyle x^{2}=x+1}
    • Supergolden ratio – the only real solution of the equation x 3 = x 2 + 1 {\displaystyle x^{3}=x^{2}+1}

References

  1. ^ Sloane, N. J. A. (ed.). "Sequence A272874". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ (sequence A137421 in the OEIS)
  3. ^ Panju, Maysum (2011). "A Systematic Construction of Almost Integers" (PDF). The Waterloo Mathematics Review. 1 (2): 35–43.
  4. ^ "Hilbert class field of a quadratic field whose class number is 3". Mathematics stack exchange. 2012. Retrieved May 1, 2024.
  5. ^ Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1.
  6. ^ Johansson, Fredrik (2021). "Modular j-invariant". Fungrim. Retrieved April 30, 2024. Table of Hilbert class polynomials
  7. ^ Weisstein, Eric W. "Elliptic integral singular value". MathWorld.
  8. ^ (sequence A008998 in the OEIS)
  9. ^ Mahon, Br. J. M.; Horadam, A. F. (1990). "Third-order diagonal functions of Pell polynomials". The Fibonacci Quarterly. 28 (1): 3–10.
  10. ^ Similar sequences are studied in: Adams, William; Shanks, Daniel (1982). "Strong Primality Tests that are Not Sufficient". Mathematics of Computation. 39 (159). American Mathematical Society: 255–300. doi:10.1090/S0025-5718-1982-0658231-9. JSTOR 2007637.
  11. ^ Analogue to the construction in: Crilly, Tony (1994). "A Supergolden Rectangle". The Mathematical Gazette. 78 (483): 320–325. doi:10.2307/3620208.