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Hi,

Many thanks in advance if you could help answering the following questions:

Q1 - Is M(e) > M(pi) true in the real numbers?

Q2 - Is the first order formal system T = ZFC + { M(e) > M(pi) } consistent?

(Where e and pi are the 2 familiar transcendental real numbers).

=========================> Definitions.

Let a real number x be generally expressed as:

I.d1d2d3...dn...

Where 'I' is the integral part and each 'dn' is a decimal expansion digit. Consider the sequence Sn defined as:

S1 = .d1d2d3...dn...

S2 = .d2d3...dn...

...

Sn = .dn(dn+1)(dn+2)...

...

Let's define M(x) ["Major number" of x] and m(x) ["minor" number of x] as:

M(x) = l.u.b (Sn)

m(x) = g.l.b (Sn)

Thanks,

-Nam Nguyen

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6 
 This is the 3rd post today by this user on the theme is this usolved problem independent of your favorite formal system. – Benjamin Steinberg Aug 23 at 16:22
May I ask on what ground that, say, Q1 is "not a real question", deserving being looked at? I've looked at the FAQ and couldn't see why, how, one could conclude Q1 is not a mathematical question about the real numbers! – Nam Nguyen Aug 23 at 18:56
FAQ stipulates that "MathOverflow's primary goal is for users to ask and answer research level math questions, ...". Unless you stipulates that there's known answer available in standard textbooks, or that this is a trivial question, it's hard for anyone to see how we can't consider Q1 as a "research level" math question. What would MO be really for, if we keep discarding genuine research level math questions? Sincerely, -Nam Nguyen – Nam Nguyen Aug 23 at 19:04
3 
 From FAQ: What about open problems? MathOverflow is not the right place to ask open problems. You should post questions you're actually seriously thinking about. If you're thinking about a well-known open problem, provide some background and ask about something specific related to the problem, like "Such and such is a well-known open problem. So-and-so proposed this and that approach in the 80s. Does anybody know if this aspect of their proposal can be made to work under these circumstances?" If you want to contribute to (or view) a list of open problems, visit the Open Problem Garden. – Benjamin Steinberg Aug 23 at 19:17
3 
 Please see my comments to the previous question mathoverflow.net/questions/105298/…. I have nothing more to add or discuss. – David Roberts Aug 24 at 0:00
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closed as not a real question by Benjamin Steinberg, Henry Cohn, Andres Caicedo, Gerald Edgar, Yemon Choi Aug 23 at 18:40

2 Answers

4

In my opinion, the answer to both questions in no, because I believe that $M(e)=M(\pi)=1$ and that this has a proof in ZFC. But, as far as I know, no such proof has been found, so the preceding sentence represents only my opinion, not a theorem.

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Hello Andreas. We answered 6 seconds apart! – Joel David Hamkins Aug 23 at 15:49
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The major number of any normal number is $1$, since one will find arbitrary long finite strings of successive $9$s in the decimal expansion, and both $e$ and $\pi$ are widely thought to be normal. Similarly, the minor number of any normal number will be $0$, since there are long sequences of $0$s in the expansion.

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But, imho, that still wouldn't enable us to answer true or false to Q1 and Q2. – Nam Nguyen Aug 23 at 15:51
I meant at least for Q1, we'd need a genuine proof. – Nam Nguyen Aug 23 at 15:53
3 
 Yes, my point is that this is a famous open question, whose status in ZFC one would seem to need to resolve in order to answer your question. – Joel David Hamkins Aug 23 at 16:00
1 
 But I am not claiming that your question is strictly equivalent to that famous normality question, since one can imagine proving that $e$ and $\pi$ both have arbitrarily long sequences of $9$s in their expansions, or merely that $M(e)=M(\pi)$, without necessarily proving that they are normal. – Joel David Hamkins Aug 23 at 16:17
1 
 The shift map on sequences is intensely studied, along with its iterations; indeed, I would say this is a fundamental example in ergodic theory, where your concept arises quite naturally. Probably it has been looked at many times. – Joel David Hamkins Aug 24 at 11:38
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