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A million digits (feh) of π

Posted 2 years ago
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NOTE: All utility functions are defined at the end and also in the attached notebook.

You can compute the first million digits of $\pi$ without printing them in < 1 second (the 1st number in the the time of computation without printing, and the printed later image is actually only a minuscule part of the whole 10^6 digits):

tim[N[Pi, 10^6]]

0.314149, 0

enter image description here

and then print the last 99:

tim[Mod[Round[10^10^6*%], 10^99]]



Far better than digits is a continued fraction:

longer[Pi, 9]

enter image description here

This expression = $\pi$



and can be freely lengthened (or shortened):


enter image description here

Notice that $e$, unlike $\pi$, has a pattern:

longer[E, 9]

enter image description here

But proving that π will never develop a pattern is one of the great unsolved problems. For faster, but non-resumable continued fractions, Mathematica has, e.g.,

ContinuedFraction[Pi, 9]

{3, 7, 15, 1, 292, 1, 1, 1, 2}

Note the largish term at position 431:

Take[ContinuedFraction[Pi, 433], -4]

{1, 4, 20776, 1}

Around 1986, in a calculation taking several weeks, I found

tim[Take[ContinuedFraction[Pi, 11504932], -4]]


{1, 1, 878783625, 6}

I thought Eric Weisstein found a bigger one, but Pi Continued Fraction page doesn't seem to say. Simple functions of e also have patterns:

longer[(E^2 + 1)/(E^2 - 1), 9]

enter image description here

$e^x$ has a nice infinite series:

SeriesCoefficient[E^x, {x, 0, n}]

enter image description here

This means

(#1 == Activate[#1] & )[Inactive[Sum][x^n/n!, {n, 0, Infinity}]]

enter image description here

Taking a few terms

ser = Activate[%[[1]] /. Infinity -> 9]

enter image description here

we can numerically test Euler's celebrated

(#1 == Activate[#1] & )[E^Inactivate[I*Pi]]

enter image description here

N[ser /. x -> I*Pi]

-0.976022 + 0.00692527 I

The square root of Euler's identity is

(#1 == Activate[#1] & )[E^(Inactivate[I*Pi]/2)]

enter image description here

N[ser /. x -> I*(Pi/2)]

0.0000247373 + 1. I

We can even use Euler's identity to calculate π by solving

eq = E^(I*x) + 1 == 0

enter image description here

This has infinitely many solutions!


enter image description here

Unprotect[C]; Table[%[[1, 1]], {C[1], -2, 2}]

{x -> -3 π, x -> -π, x -> π, x -> 3 π, x -> 5 π}

Newton's iteration says: to solve $f(x)=0$, choose an initial guess for x and iterate $g(g(...g(x)))$ where $g = x - f/ df/dx$

g[x] = Simplify[x - eq[[1]]/D[eq[[1]], x]]

enter image description here

Starting with a very precise value of 3

NestList[I/E^(I*#1) + #1 + I & , 3.`69., 7]

enter image description here

Each iteration of Newton's method typically doubles the number of correct digits. But isn't it slightly cockeyed to seek a real answer by applying a complex function to a real approximation? Suppose we just took the real part of $g$:


x + Sin[x]

(Remembering that Euler's identity generalizes to Euler's formula:)

(#1 == ComplexExpand[#1] & )[E^(I*x)]

enter image description here

Using this new $g$

NestList[#1 + Sin[#1] & , 3.`69., 5]

enter image description here

triples the accuracy each time! What if we start with 9 instead of 3?

NestList[#1 + Sin[#1] & , 9.`69., 5]

enter image description here

Dividing the last approximation by 3



Utility functions

These are definitions of all utility functions you will need for the above evaluations.




shorter[cf_,n_: 1]:=shorter[cf/.(a_: 0)+1/HoldForm[r_]:>HoldForm@@{(a+1/r)},n-1]


longer[x_?NumericQ,n_: 1]:=longer[HoldForm@@{x},n]

longer[cf_,n_: 1]:=longer[cf/.HoldForm[r:Except[_Integer]]:>

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