A counter-example to the Gelfond–Schneider theorem

[n_kelthuzad]

Gelfond–Schneider theorem can be seen here(http://en.wikipedia.org/wiki/Gelfond's_theorem) wiki.
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Given a simple calculation:
a^b
where a<0;
and let b be a fraction : u/v
so there are 3 possible ways of u and v s' arrangement:
I. u even, v odd
II. u odd, v odd
III. u odd, v even.
in I, a^b>0
in II, a^b<0
in III, a^b is a complex number.​

then the part of my study kicks in:
So what happens when b is a transcedental number?
since b is transcedental;
it no longer can be described as a fraction.
so the above 3 rules doesn't apply.
So I pick one of the most common transcedental numbers for b : b = e
and (-1) for a.
so the equation is: (-1)^e
it is known that e=1+1/1!+1/2!=1/3!+...
so it becomes: (-1)^1*(-1)^1*(-1)^(1/2)*(-1)^(1/6)...
and for all exponents starting from the 3rd one: the above (III) rule applys.
so the answer to the equation is : -1*-1*i*i*i...
=i^∞
(so this is what I was working up to now)
so there can be 2 conjectures:
1. (-1)^e = 1 or -1 or i or -i
2. the equation is undefined.
but either way, we will just leave it alone for a moment.
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If the Gelfond–Schneider theorem is true:
then (-e)^e must be transcendental.
and (-e)^e is equal to [(-1)^e]*[e^e]
still if the theorem is true; e^e must be transcendental.
then assume 1 from the above conjecture is true:
there are 2 complex answers of (-e)^e
or assume 2 is true:
(-e)^e is undefined
so clearly either way, it is a paradox.


By Victor Lu, 16
Burnside High School, Christchurch, NZ
496200691@qq.com

 

[chiro]

Gelfond–Schneider theorem can be seen here(http://en.wikipedia.org/wiki/Gelfond's_theorem) wiki.
----------------------------------------------------------------------------------

Given a simple calculation:
a^b
where a<0;
and let b be a fraction : u/v
so there are 3 possible ways of u and v s' arrangement:
I. u even, v odd
II. u odd, v odd
III. u odd, v even.
in I, a^b>0
in II, a^b<0
in III, a^b is a complex number.​

then the part of my study kicks in:
So what happens when b is a transcedental number?
since b is transcedental;
it no longer can be described as a fraction.
so the above 3 rules doesn't apply.
So I pick one of the most common transcedental numbers for b : b = e
and (-1) for a.
so the equation is: (-1)^e
it is known that e=1+1/1!+1/2!=1/3!+...
so it becomes: (-1)^1*(-1)^1*(-1)^(1/2)*(-1)^(1/6)...
and for all exponents starting from the 3rd one: the above (III) rule applys.
so the answer to the equation is : -1*-1*i*i*i...
=i^∞
(so this is what I was working up to now)
so there can be 2 conjectures:
1. (-1)^e = 1 or -1 or i or -i
2. the equation is undefined.
but either way, we will just leave it alone for a moment.
------------------------------------------------------------------------
If the Gelfond–Schneider theorem is true:
then (-e)^e must be transcendental.
and (-e)^e is equal to [(-1)^e]*[e^e]
still if the theorem is true; e^e must be transcendental.
then assume 1 from the above conjecture is true:
there are 2 complex answers of (-e)^e
or assume 2 is true:
(-e)^e is undefined
so clearly either way, it is a paradox.


By Victor Lu, 16
Burnside High School, Christchurch, NZ
496200691@qq.com

Hey n_kelthuzad and welcome to the forums.

It might help you to realize the following identity:

Consider the same c = a^b number. If a < 0 then we can write this number as:

c = a^b = |a|^b x e^(ib) where |a| is the absolute value of a and y is known as the argument of the complex number which is given by calculating:

The way you can interpret is that if you graph two functions corresponding to the real part of c and the imaginary part, then you'll get something that looks like a wave but it will either keep increasing or decreasing as the b value increases. I'm assuming also that b is a real number.

If you are having trouble, then a normal sine or cosine wave is when a = -1. If -1 < a < 0 then the peak and the trough of the wave will go towards the x-axis as b goes to infinity. If a < -1 then the peak and the trough will go towards infinity.

 

[n_kelthuzad]

Consider the same c = a^b number. If a < 0 then we can write this number as:

c = a^b = |a|^b x e^(ib) where |a| is the absolute value of a and y is known as the argument of the complex number which is given by calculating:

The way you can interpret is that if you graph two functions corresponding to the real part of c and the imaginary part, then you'll get something that looks like a wave but it will either keep increasing or decreasing as the b value increases. I'm assuming also that b is a real number.

If you are having trouble, then a normal sine or cosine wave is when a = -1. If -1 < a < 0 then the peak and the trough of the wave will go towards the x-axis as b goes to infinity. If a < -1 then the peak and the trough will go towards infinity.

c = a^b = |a|^b x e^(ib) ?

so if a= -1, b=2 c=1; then |-1|^2*e^(2i) = e^2i?
i suppose that can only be e^2i∏ which equals to 1.
and what's this formula called?

 

[chiro]

c = a^b = |a|^b x e^(ib) ?

so if a= -1, b=2 c=1; then |-1|^2*e^(2i) = e^2i?
i suppose that can only be e^2i∏ which equals to 1.
and what's this formula called?

e^(ix) = cos(x) + i sin(x) where i is the square root of -1. In other words i = SQRT(-1) and i^2 = -1.

Here is some more information:

http://en.wikipedia.org/wiki/Euler's_formula

 

[CellCoree]

Thank you for sharing your findings and conjectures. It is always important to question and examine mathematical theorems to see if there are any counter-examples or exceptions. Your example with (-1)^e and the use of e as a transcendental number is interesting and raises some valid points. However, it is important to note that the Gelfond-Schneider theorem applies to numbers of the form a^b, where a and b are algebraic numbers. In your example, e is not algebraic, so the theorem does not necessarily apply. It would be important to further explore the properties of e as a transcendental number and its interactions with other numbers in order to fully understand its behavior in equations like (-1)^e. Keep questioning and exploring, as that is the essence of science!