0010: Article 0 (Pi Formulas 1)

1/π = 3 Σ c (63n+8)/(15³)^n+1/2

1/π = 4 Σ c (154n+15)/(32³)^n+1/2

1/π = 12 Σ c (342n+25)/(96³)^n+1/2

x₁ = e^πi/24 η(τ)/η(2τ)

x₂ = e^πi/12 η(τ)/η(3τ)

x₃ = e^πi/6 η(τ)/η(5τ)

leading to,

e^π√163 ≈ x₁²⁴ - 24.0000000000000010…

e^π√163 ≈ x₂¹² - 12.00000000000000021…

e^π√163 ≈ x₃⁶ - 6.000000000000000034…

where the eta quotients x_i are the algebraic numbers given above. There are also nice Diophantine relationships for the approximants of degree 1,

7 (12³ + 15³) = 3² *63²

11 (12³ + 32³) = 4² *154²

19 (12³ + 96³) = 12² *342²

43 (12³ + 960³) = 12² *16254²

67 (12³ + 5280³) = 12² *261702²

163 (12³ + 640320³) = 12² *545140134²

So the sums are perfect squares. And where else do the blue and red numbers appear? Of all places, in formulas for π. Let c = (-1)ⁿ(6n)! / ((n!)³(3n)!), then,

Pi Formulas

I. Class number h(-d) = 1

II. Class number h(-d) = 2, where d = 4m

III. Class number h(-d) = 2, where d = 3m

(For an updated summary, also see Ramanujan's Pi Formulas and the Hypergeometric Function.)

I. Class number h(-d) = 1. It is quite well-known that,

e^π√7 ≈ 15³ + 697

e^π√11 ≈ 32³ + 738

e^π√19 ≈ 96³ + 743

e^π√43 ≈ 960³ + 743.999…

e^π√67 ≈ 5280³ + 743.99999…

e^π√163 ≈ 640320³ + 743.99999999999…

with the last also known as the Ramanujan constant. The discriminants d under the square root are the six highest of the nine Heegner numbers. As I pointed out in a 2008 sci.math.research post, it is not so commonly known there is another internal structure,

e^π√7 ≈ 3³(3²-2²)³ + 697

e^π√11 ≈ 4³(3²-1)³ + 738

e^π√19 ≈ 12³(3²-1)³ + 743

e^π√43 ≈ 12³(9²-1)³ + 744

e^π√67 ≈ 12³(21²-1)³ + 744

e^π√163 ≈ 12³(231²-1)³ + 744

The reason for the squares of {3, 9, 21, 231} is due to a certain Eisenstein series, and they will appear again below. Before going to the pi formulas, it can be pointed out that these transcendental numbers, in addition to being closely approximated by integers (which are simply algebraic numbers of degree 1), can also be closely approximated by algebraic numbers of degree 3,

e^π√11 ≈ x²⁴ – 24; (x³-2x²+2x-2 = 0)

e^π√19 ≈ x²⁴ – 24; (x³-2x-2 = 0)

e^π√43 ≈ x²⁴ – 24; (x³-2x²-2 = 0)

e^π√67 ≈ x²⁴ – 24; (x³-2x²-2x-2 = 0)

e^π√163 ≈ x²⁴ – 24; (x³-6x²+4x-2 = 0)

The corresponding cubic equations are extremely simple and have a very similar form. In fact, x can be exactly expressed in terms of the Dedekind eta function η(τ), a modular function which involves a 24th root, and explains the 24 in the approximations. (Note: As Rich Schroeppel points out, John Brillhart, in the early days of computers, looked at continued fractions of cubic irrationalities and found a few had some interesting properties, one of which was x³-6x²+4x-2 = 0. If you look at the first 200 terms, it is curiously peppered with large terms. Of course, the fact that it is related to e^π√163 gives an inkling of the reason why.)

These transcendental numbers can also be closely approximated by algebraic numbers of degree 4,

e^π√19 ≈ 3⁵(3-√(2v₁))^-2 - 12.00006…

e^π√43 ≈ 3⁵(9-√(2v₂))^-2 - 12.000000061…

e^π√67 ≈ 3⁵(21-√(2v₃))^-2 - 12.00000000036…

e^π√163 ≈ 3⁵(231-√(2v₄))^-2 - 12.00000000000000021…

(notice how the numbers {3, 9, 21, 231} appear again) and where the v_i are,

v₁ = -3 + 1√(3*19)

v₂ = -39 + 7√(3*43)

v₃ = -219 + 31√(3*67)

v₄ = -26679 + 2413√(3*163)

and that,

3*2⁶(-3² + 3*19*1²) = 96²

3*2⁶(-39² + 3*43*7²) = 960²

3*2⁶(-219² + 3*67*31²) = 5280²

3*2⁶(-26679² + 3*163*2413²) = 640320²

which, with the appropriate fractional power, are precisely the j-invariants. As well as for algebraic numbers of degree 6,

e^π√19 ≈ (5x)³ - 6.000010…

e^π√43 ≈ (5x)³ - 6.000000010…

e^π√67 ≈ (5x)³ - 6.000000000061…

e^π√163 ≈ (5x)³ - 6.000000000000000034…

where the x’s are given respectively by the appropriate root of the sextics,

5x⁶-96x⁵-10x³+1 = 0

5x⁶-960x⁵-10x³+1 = 0

5x⁶-5280x⁵-10x³+1 = 0

5x⁶-640320x⁵-10x³+1 = 0

with the j-invariants appearing again. Another interesting feature of these sextics is that they are not only algebraic, but are also solvable in radicals. These factor into two cubics (with the exception of the first which has a quadratic factor) over the extension Q(φ) where φ = (1+√5)/2 or the golden ratio. The last three, respectively, then have the cubic factors,

5x³ - 5(53+86φ)x² + 5(8+13φ)x - (18+29φ) = 0

5x³ - 20(73+118φ)x² - 20(21+34φ)x - (47+76φ) = 0

5x³ - 20(8849+14318φ)x² + 20(377+610φ)x - (843+1364φ) = 0

Amazingly, pairs of consecutive Fibonacci numbers F(n) = {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,…} appear in the x term, as well as Lucas numbers L(n) = {2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364,…} in the constant term. Why this is so I have no idea as it came as a surprise while I was looking at their factorization over Q(φ). If you have Mathematica, the factorization is easily verified by the Resultant[] function,

Resultant[5x³ - 5(53+86φ)x² + 5(8+13φ)x - (18+29φ), φ²-φ-1, φ]

which eliminates the variable φ between the two equations and recovers the original sextic. (Likewise for the other two.) (Update, 5/23/11) Qiaochu Yuan from MathStackExchange gave the identity for Fibonacci numbers,

F_n-1 + F_n φ = φⁿ

A related one for Lucas numbers is,

L_n-1 + L_n φ = φⁿ√5

Hence, the cubic factors simplify as,

5x³ - 5 (4φ⁸ + φ³) x² + 5φ⁷ x - φ⁷√5 = 0

5x³ - 20 (2φ¹⁰ + φ⁶) x² - 20φ⁹ x - φ⁹√5 = 0

5x³ - 20 (4√5 φ¹⁷ + φ⁹) x² + 20φ¹⁵ x - φ¹⁵√5 = 0

or the coefficients neatly contain powers of φ. (End update) These algebraic approximants of degree 3,4,6 can be exactly expressed as Dedekind eta η(τ) quotients. For example, let τ = (1+√-163)/2, then,