My Math Forum Transcendence of Gelfond's-like Constants

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 May 4th, 2013, 03:45 AM #1 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Transcendence of Gelfond's-like Constants Can it be proved whether $\mathbb{e}^{\pi + \alpha}$ for $\alpha \in \mathbb{Q}$ is transcendental or not? It doesn't seem that it can be deduced from any known results; even I think it can't be proved conditionally using the current conjectures on transcendence theory. Balarka .
 May 7th, 2013, 11:21 AM #2 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Transcendence of Gelfond's-like Constants Anyone?
 May 7th, 2013, 02:34 PM #3 Senior Member   Joined: Mar 2012 Posts: 572 Thanks: 26 Re: Transcendence of Gelfond's-like Constants I only wish I could even pretend to understand the question.
May 7th, 2013, 03:04 PM   #4
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Re: Transcendence of Gelfond's-like Constants

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 Originally Posted by Hedge I only wish I could even pretend to understand the question.
Me too!

May 7th, 2013, 03:09 PM   #5
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Re: Transcendence of Gelfond's-like Constants

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 Originally Posted by Hedge I only wish I could even pretend to understand the question.
Its negation is: "Is there some rational number a and nonzero integer polynomial P(x) such that P(exp(Pi + a)) = 0?".

I don't think that the question can be answered at present.

May 7th, 2013, 10:18 PM   #6
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Re: Transcendence of Gelfond's-like Constants

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 Originally Posted by Hedge I only wish I could even pretend to understand the question.
See this link : http://en.wikipedia.org/wiki/Transcendental_number. The explanation CRGreathouse gave is concise and elegant when compared to a article, I'd say.

May 8th, 2013, 06:16 AM   #7
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Re: Transcendence of Gelfond's-like Constants

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Originally Posted by CRGreathouse
Quote:
 Originally Posted by Hedge I only wish I could even pretend to understand the question.
Its negation is: "Is there some rational number a and nonzero integer polynomial P(x) such that P(exp(Pi + a)) = 0?".

I don't think that the question can be answered at present.
Thanks, I can't totally picture but at least that helps me understand what kind of question it is.

May 8th, 2013, 06:37 AM   #8
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Re: Transcendence of Gelfond's-like Constants

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 Originally Posted by Hedge Thanks, I can't totally picture but at least that helps me understand what kind of question it is.
Let me try to add an overview, then. There are a small number of 'nice' numbers which can be expressed as the roots of an integer polynomial. These numbers are called algebraic, and they include the integers and the rationals since a/b is the solution to bx - a = 0. But they also include numbers like the golden ratio, the square root of two, and the real root of x^5 - x - 1 (which cannot be expressed as a closed-form expression with field operations plus root extraction).

There are numbers we know not to be algebraic, like e. We call such numbers "transcendental". As it happens we know that e^Pi is also transcendental. The question is whether we can extend the proof that e^Pi is transcendental to numbers of the form e^(Pi + a) for a rational, or else find a different proof that such a number is transcendental. I don't know but suspect this is beyond current capabilities.

 May 8th, 2013, 07:00 AM #9 Math Team   Joined: Apr 2012 Posts: 1,579 Thanks: 22 Re: Transcendence of Gelfond's-like Constants Yes, I was quite surprised when I first found out that some irrationals behave in certain significant ways the way rationals do and unlike the way certain other fellow irrationals do. Are there any other significant subdivisions of irrationals like this algebraic vs transcendental distinction?
May 8th, 2013, 10:07 AM   #10
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Re: Transcendence of Gelfond's-like Constants

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 Originally Posted by johnr I was quite surprised when I first found out that some irrationals behave in certain significant ways the way rationals do and unlike the way certain other fellow irrationals do.
It's surprising that you didn't knew about transcendental numbers before . . .

Quote:
 Originally Posted by johnr Are there any other significant subdivisions of irrationals like this algebraic vs transcendental distinction?
I don't think there are any (remarkable or useful or notable) such examples, although I am not so sure about it either.

Quote:
 Originally Posted by CRGreathouse The question is whether we can extend the proof that e^Pi is transcendental to numbers of the form e^(Pi + a) for a rational
This can be extended further for algebraic a. One thing we know that at least one of e^Pi + e^a and e^(Pi + 1) is transcendental but probably this fact is of no use since it's reasonable to guess that both are transcendental.

What is the irrationality measure of exp(Pi+a) for rational a? Forget the a, what is the irrationality measure for exp(Pi) ?

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