- #1

n_kelthuzad

- 26

- 0

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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: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.

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