my hypothesis/proof that pi is a normal number

Y'er a bit late. It certainly appears that pi is a normal number, not only in base 2 and in base 10 but in other bases as well. But sadly, as noted above, there is no proof. (Some very recent empirical analysis of the first 16 trillion binary digits, corresponding to a bit less than 5 trillion decimal digits -- by us and several co-authors -- strongly suggests that pi is normal base 2, but again this is not a proof.) So, all you have to do is compute Pi to a zillion digits in a zillion bases, run the digit distribution, and you'll get your Ig Nobel Prize for wasting everyones time and for posting an off topic message without clearly labeling it as OT.

I am not a mathematician by any means, so please advise how PI may be calculated to such a large number of digits. My guess is that it is approximated by determining the circumference of a circle as a number of isosceles triangles with legs = radius and adding the lengths of the bases will approach PI as the number => infinity.

But on a practical basis, there is a limit to the precision of any such physical measurement, perhaps based on the Planck Constant

Paul

Usually using some variation of the Machin formula:

This gives you more than one new decimal digit per iteration, because the slowest converging term is the odd powers of 1/5.

There is no such limit to the calculation of the theoretical ratio c/d. No physical measurement is required.

Clifford Heath.

Well, Pi is just "1" in base Pi. Numerical bases are nothing more than an interface to physical storage - stones, check marks, fingers, bits, trellis codes, flash cell levels, etc. That's why Pi remains in symbolic form as formulas are processed.

Hi,

Pi can be expressed as various continued fractions or infinite series:

There are obvious fractal patterns just in the infinite series themselves that are what pi is. Although the digits of pi look random there is some underlying order related to these infinite series and/or continued fractions too I guess.

cheers, Jamie

So I wonder why anyone would want to calculate PI to such extreme precision. I just tried my method of calculating:

PI = n * sin(180/n)

n f(n)

Using PI = n * tan(180/n)

n f(n)

As expected, the values converge from below and above using sin() or tan()

The average of the values using 36000 is:

This differs from PI by:

Paul

Sounds like there is a false presumption somewhere...

David Bailey! Years ago i used some of his multi-million digit math algorithms to write my own My Dear Aunt Sally in Fortran. Allocate/deallocate large blocks of memory as needed without losing a single byte (AKA "memory leak"). He analyzed how to program groups of multiplies to avoid "burps" caused by cache refilling at inappropriate times. He is the computer programing master on how-system-works and work-arounds.

On Sun, 01 Nov 2015 14:32:25 -0800, Jeff Liebermann Gave us:

That would be the "Usenet Ignoble Prize", no?

The only hypothesis these "digit hunters" are proving is that observation of huge amounts of data is not a substitute for theoretical insight.

Hi,

Ya I think so, but I have found a difference in how pi compresses versus all the other sequences I've tested, up to 60million digits, so I'm just tracking down what that difference is exactly now.

The compression difference I am getting is most likely due to one or more of these things or a combination of them (I can find out for sure after some more checking)

1. the digits of pi have fewer "local" (ie within 1million consecutive digits occuring anywhere up to 60million digits) matched repeating pairs of digits ie 3,4 or 5,6 repeated at least twice.

1. the digits of pi have fewer "local" (ie within 1million consecutive digits occuring anywhere up to 60million digits) matched repeating pairs of digits with equal spacing occuring at least three times ie.

3,4,x,y,z,3,4,a,b,c,3,4 (matched spacing gap of 3 digits between 3,4)

Or could be something else or related to the above at least for the difference I think I found in pi.

cheers, Jamie