# My Quibbles

The Inverse Symbolic Calculator 1 and Plouffe's Inverter 2 are great resources, but they failed to satisfy me, primarily for the following reasons:

Okay, enough complaining. Back to ries main page

### Glossary for ISC and Plouffe's Inverter

I got tired of complaining and eventually started a list of some of the constants and functions used in ISC output.

Artin

Artin's constant 0.3739558136... (see Artin's conjecture on primitive roots)

example:

2506186020926769 = (0001) GAM(3/4)+Artin+GAM(1/24)

{~=} 1.2254167024 + 0.3739558136 + 23.462487693

BesK

The modified Bessel function of the second kind, commonly represented by the symbol Kα(x) for some α and x. In Maple and Mathematica this is written BesselK(α,x).

example:

6019072301972345 = (0001) BesK(1,1)

6019072301972345 = (0297) BesselK(1,1)

BesselK

Synonym for BesK.

Catalan

Catalan's constant 0.915965594177219...

example:

2506144130719856 = (0001) Catalan*(sr(5)+1/2)

{~=} 0.915965594177 * (2.2360679774998 + 0.5)

exp(...)

The exponential function

example:

|1535063009255209 = (0028) exp(x) x=3/7

F(...)

The Hypergeometric function with Maple syntax

examples:

1535063009255209 = (0231) F(1,1;3/7)

1535063009255209 = (0226) F(1/3,1/3;3/7)

2506169291781055 = (0249) F(10/23;19/39;1)

2506179399171396 = (0157) F(2/11,10/11;1/2;9/10)

Feig1

The first of the two Feigenbaum constants, 4.669201609102990...

examples:

4669201609102990 = (0000) Feigenbaum constant

4669201609102990 = (0001) Feig1

|1535061093537132 = (0001) exp(-Pi)/(K(1/sr(2))-Feig1)

GAM(...)

examples:

2506186020926769 = (0001) GAM(3/4)+Artin+GAM(1/24)

{~=} 1.2254167024 + 0.3739558136 + 23.462487693

3625609908221908 = (0001) GAM(1/4)

3625609908221908 = (0001) Pi*sr(2)/GAM(3/4)

Gamma(...)

The Gamma function

example:

3625609908221908 = (0092) (Gamma(1/4))^n n=1

GAMMA(...)

The Gamma function

examples:

3625609908221908 = (0000) GAMMA(1/4)

3625609908221908 = (0032) GAMMA(1/4)

K(...)

The Complete Elliptic Integral of the first kind. In ISC the argument is squared; for example, "K(1/sqrt(2))" in ISC meansEllipticK[1/2] using the Mathematica EllipticK[] function, and is equal to 1.854074677301...

In general, ISC's "K(x)" is equal to EllipticK[x^2] inMathematica. Using PARI/GP, ISC's "K(x)" is equal to "(Pi/2) /agm(1-x,1+x)" where we are computing the elliptic integral via the arithmetic–geometric mean.

examples:

2506185982626307 = (0001) K(1/sr(2))/(sin(Pi/5)^W(1))

2085935113295952 = (0001) K(1/sr(2))^GAM(1/3)-Pi

2085941674074939 = (0001) 2/3+K(1/sr(2))^W(1)

example:

Parking

Renyi's Parking constant 0.7475979202534...

example:

roots(...)

One or more roots of a polynomial

example:

1535063009255209 = (0276) exp(roots(-7*x^2-4*x+3))

= exp(3/7)   (because 3/7 is a solution of -7*x^2-4*x+3=0)

sqrt(..)

Square root

example:

2506147722382783 = (0007) sqrt((21+sqrt(17))/4)

{~=} sqrt((21 + 4.1231056256)/4)

sr(..)

Square root

example:

2506144130719856 = (0001) Catalan*(sr(5)+1/2)

{~=} 0.915965594177 * (2.2360679774998 + 0.5)

sum(...)

Sum of an infinite series, as normally indicated with "big Sigma" notation

examples:

2506181684414645 = (0013) sum((2/3*n^3-3/2*n^2+47/6*n+4)/n^(n-1),n=1..inf)

2506185297338450 = (0232) sum(1/(n!*phi(5,7,3)),n=1..inf)

TwinPrim

The Twin prime constant, 0.6601618158...

example:

2506141791780115 = (0001) (TwinPrim+exp(sr(2)))^sin(Pi/5)

{~=} (0.6601618158 + e^1.414213562) ^ sin(0.5877852522)

W(...)

example:

5671432904097838 = (0000) W(1)

x

An ISC result consisting of a polynomial in x represents a root of that polynomial (a value that x needs to have for the polynomial's value to be zero).

example:

6019074146223354 = (0315) -2+x+2*x^2+5*x^4+x^8

(because if x {~=} 0.6019074146223354..., then x8 + 5x4 + 2x2 + x - 2 is zero)

Sources

Plouffe's Inverter was formerly here, but is now (as of mid 2013) too big to have online but is still maintained by its creator Simon Plouffe.

Footnotes

1 : Plouffe's Inverter has partly addressed the progressive complexity issue by sorting its output, but it continues to only report matches that agree in the digits given by the user. Also, it scores its matches purely by string length when rendered into ASCII, so for example "Riemannzero1" gets the same score as "PisotV*Feig2" and "820421/78823" regardless of which is most exceptional.

2 : Just for fun, here are the results ries finds for the natural log of successive Fibonacci ratios:

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2020 Mar 26. s.11