## Sunday, August 27, 2017

### either pi is algebraic or some journals let in an incorrect paper!/the 15 most famous transcendental numbers

Someone has published three papers claiming that

π is 17 -8*sqrt(3) which is really =3.1435935394...

Someone else has published eight papers claiming

π is (14 - sqrt(2))/4 which is really 3.1464466094...

The first result is closer, though I don't think this is a contest that either author can win.

Either π is algebraic, which contradicts a well known theorem, or some journals accepted some papers with false proofs. I also wonder how someone could publish the same result 3 or 8 times.

I could write more on this, but another blogger has done a great job, so I'll point to it: here

DIFFERENT TOPIC (related?) What are the 15 most famous transcendental numbers? While its a matter of opinion, there is an awesome website that claims to have the answer here. I"ll briefly comment on them. Note that some of them are conjectured to be trans but have not been proven to be. So should be called 12 most famous trans numbers and 3 famous numbers conjectured to be trans. That is a bit long (and as short as it is only because I use `trans') so the original author is right to use the title used.

1) pi YEAH (This is probably the only number on the list such that a government tried to legally declare its value, see here for the full story.)

2) e YEAH

3) Eulers contant γ which is the limit of (sum_{i=1}^n  1/i) -  ln(n). I read a book on γ  (see here) which had interesting history and math in it, but not that much about γ . I'm not convinced the number is that interesting. Also, not known to be trans (the website does point this out)

4) Catalan's number  1- 1/9 + 1/25 - 1/49 + 1/81  ...  Not known to be trans but thought to be. I had never heard of it until reading the website so either (a) its not that famous, or (b) I am undereducated.

5) Liouville's number 0.110001... (1 at the 1st, 2nd, 6th, 24th, 120th, etc - all n!- place, 0's elsewhere)
This is a nice one since the proof that its trans is elementary. First number ever proven Trans. Proved by the man whose name is on the number.

6) Chaitian's constant which is the prob that a random TM will halt. See here for more rigor. Easy to show its not computable, which implies trans.  It IS famous.

7) Chapernowne's number which is 0.123456789 10 11 12 13 ... . Cute!

8) recall that ζ(2) = 1 + 1/4 + 1/9 + 1/6 + ... = π2/6

ζ(3) = 1 + 1/8 + 1/27 + 1/64 + ... known as Apery's constant, thought to be trans but not known.

It comes up in Physics and in the analysis of random minimal spanning trees, see here which may be why this sum is here rather than some other sums.

9) ln(2). Not sure why this is any more famous than ln(3) or other such numbers

10) 2sqrt(2) - In the year 1900 Hilbert proposed 23 problems for mathematicians to work on (see here for the problems, and see here  for a joint book review of two books about the problem, see  here for a 24th problem found in his notes much later). The 7th problem  was to show that ab is trans when a is rational and b is irrational (except in trivial cases). It was proven by Gelfond and refined by Schneider (see here). The number 2sqrt(2) is sometimes called Hilbert's Number. Not sure why its not called the Gelfond-Schneider number. Too many syllables?

11) eπ  Didn't know this one. Now I do!

12) πe (I had a post about comparing eπ to πe  here.)

13) Prouhet-Thue-Morse constant - see here

14) i^i. Delightful! Its real and trans! Is it easy to show that its real? I doubt its easy to show that its trans. Very few numbers are easy to show are trans, though its easy to show that most numbers are.

15) Feigenbaum's constant- see here

Are there any Trans numbers of which you are quite fond that aren't on the list?

If you proof any of the above algebraic then you can probably publish it 3 or 8 or ii times!

I can imagine a crank trying to show π or maybe even e is algebraic. ζ(3) or the Feigenbaum's constant, not so much.

1. What is the definition of i^i? It seems that you have to give up some property of the power function to properly define it. For example i^5=i, so if we define t=i^i, then t^5 = i^{5i}=(i^5)^i = i^i = t, which in particular means that t is algebraic. If you want to remain consistent with Euler's formula then you're in bigger trouble since i=exp(i*(pi/2+2pi*k) for any integer k, raise this to the power i and you get an inconsistent value.

2. "(Actually, this is one of many possible values for i to the i, because, for instance, exp(5i*Pi/2)=i. In complex analysis, one learns that exponentiation with respect to i is a multi-valued function.)"

From: https://www.math.hmc.edu/funfacts/ffiles/20013.3.shtml

3. I think it's impossible to not be convinced that γ is interesting. Consider f(n)=log(n!), and g(n)=1+1/2+...+1/n. Take the derivative of f at any point (with n! understood in terms of gamma function). The difference between the result and g(n-1) is ALWAYS equal to γ. Euler's constant is all over the place in the theory of special functions. It's arguably as fundamental as e or pi.

4. i^i = exp(i Log(i)) = exp(i(log(1) + iArg(i))) = exp(-pi/2) = exp(pi)^(-1/2) is transcendental by 11.

5. Log(2) is famous because it is the sum of the alternating harmonic series.

6. The golden ratio phi from Fibonacci numbers, which should probably take the third position for its importance: https://en.wikipedia.org/wiki/Golden_ratio

7. Sorry, abort the previous comment :)
Phi is irrational but not transcendental...