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:
| Plouffe or ISC | ries |
| Cryptic output and no glossary (but see below). Gives answers like -1+x^5+x^6-x^7-x^8+x^9+x^11+x^12 without defining what x is. Also gives many answers that can't be checked unless you own a copy of Mathematica or Maple. | All solutions are self-explanatory. Lesser-known symbols like phi are defined by a tagline in the output the first time they are used. |
| Presents all answers in numerical sequence and without regard for the complexity of the description of each number.1 | Gives answers in order of increasing complexity and increasing closeness to your number — just like continued fractions. |
| Isn't useful for people who only know a number to within 6 or 7 digits, but know that it probably has a simple formula — because there are just so many long, complex formulas to wade through. | Immediately gives you the simplest formulas or equations for any number regardless of the number of digits in your number. It also gives matches that are equal to your value in the first 6 or 7 digits, but you're free to just ignore those matches. |
| Thinks decimal points don't matter — search for 2.5061841 and it gives ln(26263/20441) = 0.25061841... as an answer! Why?!?!? | Doesn't consider "a factor of 10" to be any more important than "a factor of 9" or "a factor of 11". Numbers are weighted by size and how easily they can be formed from other numbers (giving composites an advantage). |
| As of early 2021, the ISC still didn't know that 2.506184145588... is the solution of xx = 10 (Try it) All solutions are in the form K = expression, and apparently the Lambert W function is not used when looking for this number (though I later forgot all about this, and found a Labmert W function when I was compiling the ISC Glossary below). | Reports this as the 4th solution for any value within +- 0.00044 of 2.506184. |
| Database is cluttered with stuff like log(26263/20441) and cos(Pi*23/56) / cos(Pi*15/34). | Reports things like log(441/263) only if your number is closer to that value than to the value of any simpler expression.2 |
| Doesn't let you specify "please ignore these weird functions". | Lets you specify any combination of functions, operators, and small integers to be used in a search. |
| Requires Internet connection; puts load on someone else's computer. | Runs on your computer, but fast enough to try billions of possible solutions in less than a minute. |
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(...)
The Gamma function
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)
Gam(...)
The Gamma function
gamma
The Euler-Maccheroni constant
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 means EllipticK[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] in Mathematica. 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)
Madelung
The Madelung constant -1.747564594633...
example:
2506183101609269 (0001) cos(Pi/12)-ln(1+sr(2))*Madelung
Parking
Renyi's Parking constant 0.7475979202534...
example:
2506185338545284 (0001) (W(1)-Parking*Madelung)/Parking
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
The Inverse Symbolic Calculator (with a 1997 version of the database) is available here (formerly at oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html ).
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 2023 Jul 20.
s.27