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#1 2018-07-26 11:03:10

!nval!d_us3rnam3
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Registered: 2017-03-18
Posts: 25

Find all real solutions: Proof help

Find all real solutions for

I could use detailed help and/or a solution, the sooner the better.
The only work I have is simplifying and proving that 2^x - 1 and x have the same sign.
Thanks!
!nval!d_us3rnam3

Last edited by !nval!d_us3rnam3 (2018-07-27 03:25:21)


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#2 2018-07-26 23:44:22

Alg Num Theory
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Registered: 2017-11-24
Posts: 338
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Re: Find all real solutions: Proof help

There are precisely three solutions:

Unfortunately I do not know how to prove it rigorously. Maybe Bob Bundy can come up with more helpful information.

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#3 2018-07-27 02:28:33

!nval!d_us3rnam3
Member
Registered: 2017-03-18
Posts: 25

Re: Find all real solutions: Proof help

Preferably today.
Can you give me some pointers, at least? How you solved this.


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#4 2018-07-27 09:50:57

zetafunc
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Re: Find all real solutions: Proof help

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#5 2018-07-27 11:02:13

!nval!d_us3rnam3
Member
Registered: 2017-03-18
Posts: 25

Re: Find all real solutions: Proof help

Ok, but I wasn't looking for a proof with the answers in the beginning. I've gotten pretty far, but I need to find all the answers to


EDIT: With a rigorous demonstration that this is the answer.

Last edited by !nval!d_us3rnam3 (2018-07-27 11:02:45)


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#6 2018-07-27 21:23:51

zetafunc
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Re: Find all real solutions: Proof help

!nval!d_us3rnam3 wrote:

Ok, but I wasn't looking for a proof with the answers in the beginning. I've gotten pretty far, but I need to find all the answers to


EDIT: With a rigorous demonstration that this is the answer.

That depends on what you consider to be a 'rigorous demonstration'. Clearly
is a factor, since both the LHS and RHS vanish. Dividing your equation by
(taking
) gives you

Since
vanishes whenever
, then the numerator of the term on the RHS also vanishes for these values of
. That's three solutions, and you can 'rigorously demonstrate' that there aren't any more solutions over
via the method in post #4.

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#7 2018-07-28 02:29:13

Alg Num Theory
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Registered: 2017-11-24
Posts: 338
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Re: Find all real solutions: Proof help

Let us apply zetafunc’s method in another way. Dividing by 2 and rearranging gives

Now consider the following table:

Thus we see that outside of {0,±1} the equation has no solution as the LHS and RHS have opposite signs. This proves that there are no solutions other than x = 0, ±1.

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