## Keen Approximations

Ed Pegg Jr., February 14, 2005

2 + 310×109 = 235 is a keen sum. One can measure the keenness using the techniques given at Abderrahmane Nitaj's ABC Conjecture tables. In this case, the keenness of the sum is 1.62991, which is higher than any other known sum.

Eric Weisstein has compiled many approximations under Almost Integer, Pi Approximations, Feigenbaum Constant Approximations, Euler-Mascheroni Constant Approximations, e Approximations, Khinchin's Constant Approximations, Catalan's Constant Approximations, and Apéry's Constant Approximations. For the Plouffe Invertor, Simon Plouffe compiled a few approximations of pi. Other lists can be found at Math Magic, Contest Center and Prime Puzzles and Problems Connection. There are a lot of approximations in these pages. How can I calculate which has the highest keenness?

How should Sin[11] = -0.99999020... be measured? How about the fourth root of 9.1, with a continued fraction that includes 75656, 136181, 3602, 4267, 4841, and 262344 within the first 60 terms? I considered weird, but nothing particularly special. NJA Sloane found it to be interesting enough to add it to the OEIS as A093876, due to the many unusual large terms.

For almost all numbers, the geometric mean of the continued fraction approaches Khinchin's constant. The probability that a particular part of the continued fraction is of size s is Log[2, (1 + 1/s)/(1 + 1/(s+1))] or about 1/s2 (Wolfram, p 914). Having 102108 as the third term of Sin[11] is can be considered unusual.

In asymptotic numbers, on the other hand, it's the norm. For instance, the second term of Zeta[99] is 633825300114114698409864795031, a 30 digit number. The second term of Zeta[99] is a116625189 digit number. Richard Sabey's spectacular approximation to e, (1+9-47*6)3285, gets 18457734525360901453873570 decimal digits of accuracy due to high exponents and asymptotic effects. More bizarre behavior can be found in objects like π-tan-1(1010), the continued fraction for which begins: {0, 5000000000, 60000000000, 6250000000, 62222222222, 4, 2, 78124999, 1, 1, 3, 1, 62577777777, 6347656250, 62693877551, 48, 1, 2646931, 1, 4, 2, 6, 2, 1, 3002469135, 8, 2, 1, 2, 6, 1, 17, 2, 403, 5, 2, 1, 1, 13, 1, 1, 1, 2, 3, 9199061319, ... }.

Continued fractions themselves are asymptotic. The second term in the continued fractions of 355/113 - π is 3748629. The convergents (close fractions) of π are 3, 22/7 (Archimedes), 333/106, 355/113 (Tsu Ch'ung-Chi, 480 AD), 103993/33102, and so on. If you use enough digits, you can get arbitrarily close with a fraction, or with a decimal. 3.1415926535897932385 - π has a second term of 26769019461318409709, for example.

Cleverness with functions can cause other problem. Let j = 2, nested under 355 Square Root symbols. Let k = 2, nested under 113 Square Root symbols. Then log2(j)/log2(k)=355/113 -- the same fraction above. Functions cannot be considered free when nesting can produce an infinite number of values.

I've decided keenness = log10(max(CF))/complexity. The top part, log10(max(CF)), is equivalent to the digits of accuracy. The bottom part is equivalent to the digits used, in something simple like a fraction.

My rules to measure the keenness of an approximation are as follows:

1. Each digit or constant used adds 1 to the complexity. Each function adds 1 to the complexity.
2. Division, multiplication, addition, subtraction, exponentiation, parenthesization are free. In 1/x, or in x-1, the 1 is free.
3. Asymptotic functions, such as Zeta or cos(1+cos(1+cos(x))), are not allowed, nor are Pisot numbers.
4. Root objects, such as Pi ~ {1+x+6x2+x3-5x4-5x5+2x6}2 can be thought of as {1,1,6,1,-5,-5,2}2 (complexity 8). Zeros count.

The approximations with the highest keenness are as follows (TeX):

The first of these is a recent discovery by Michael Trott. The large continued fraction term is 46658703216579428. Calculating log10(46658703216579428)/7 gives 2.381.

For the second, John Conway said of a different form: "eπ - π = 19.999099979... . I have often wondered if there's any simple explanation for the form of this number."

The next four are all variations of very deep mathematics. The sixth, the simple polynomial x3 - 6x2 + 4x - 2, sparked my interest in this topic. I first learned of it in Titus Piezas' article, Ramanujan’s Constant (eπ√163) And Its Cousins. More of the mathematics involved can be seen through the j-Function, the Ramanujan Constant, and the Pisot numbers.

262537412640768743.9999999999992500726 is eπ√163 -- a result of the j-Function
262537412640768743.9999999999992511239 is (x3 - 6x2 + 4x - 2)24 - 24. (The "-24" isn't counted in the complexity.)
262537412640768744 is 6403203 + 744.
j(q) = 1/q + 744 + (1 + 196883)q + (1+196883+21296876)q2 + (2×1+2×196883+21296876+842609326)q3 -- the j-Function
1, 196883, 21296876, 842609326, 18538750076 -- A001379 -- Degrees of irreducible representations of Monster group M
x11+2x10-x9-x7-x6-x5+2x4+x-1 (mod 4) = g4(x). With zerosum check, a self-dual code of length 24. Generates the Leech Lattice.

The Leech Lattice, connected to perfect hypersphere-packings in the 24th dimension, is seemingly connected to (x3 - 6x2 + 4x - 2)24 - 24 and eπ√163. When John McKay noticed (1+196883) from the Griess algebra and from the j-function, John Conway initially dismissed it as "moonshine." The connection was proven in 1992 by Richard Borcherds via the no-ghost theorem from string theory, which garnered him the Fields Medal, the highest honor in Mathematics.

The last four approximations seem to be lucky finds. But they might be much more than that.

If you can find other approximations with a keenness above 1.3, I'd love to hear from you (ed@mathpuzzle.com). I will maintain this list and hone the rules until a really solid top twenty list is produced. I'm curious about anything over 1.1. For example, (23/9)5 is an approximation to 109, with a keenness of 1.118. That's the best entry in the ABC Conjecture tables, and it doesn't crack the top ten.

References:

John Conway and Neil Sloane, Sphere Packings, Lattices, and Groups, Springer 1999.

Erich Friedman, Math Magic August 2004: Approximations. http://www.stetson.edu/%7Eefriedma/mathmagic/0804.html.

Titus Piezas, Ramanujan’s Constant (eπ√163) And Its Cousins. http://www.geocities.com/titus_piezas/Ramanujan_a.htm.

Simon Plouffe, Plouffe's Invertor, http://pi.lacim.uqam.ca/eng/.

Carlos Rivera, "The best approximation to pi with primes," http://www.primepuzzles.net/puzzles/puzz_050.htm.

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, page 914. http://www.wolframscience.com/nksonline/page-914d-text.

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