Find all real solutions: Proof help

!nval!d_us3rnam3 · 2018-07-26 11:03:10

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)

Alg Num Theory · 2018-07-26 23:44:22

There are precisely three solutions:

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

!nval!d_us3rnam3 · 2018-07-27 02:28:33

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

zetafunc · 2018-07-27 09:50:57

!nval!d_us3rnam3 · 2018-07-27 11:02:13

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)

zetafunc · 2018-07-27 21:23:51

!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.

Alg Num Theory · 2018-07-28 02:29:13

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

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Now consider the following table:

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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.

Math Is Fun Forum

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

Find all real solutions: Proof help

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

Re: Find all real solutions: Proof help

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

Re: Find all real solutions: Proof help

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

Re: Find all real solutions: Proof help

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

Re: Find all real solutions: Proof help

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

Re: Find all real solutions: Proof help

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

Re: Find all real solutions: Proof help

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