Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

Solve for positive real x

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Solve for positive real x

x = an integer one less than a square would be my guess.

I've proven that any solution must be an integer and that LHS ≥ RHS for any integer x, if that helps anyone else.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Solve for positive real x

If we define:

Then what we are asking is where in this function is there a fixed point.  There are various theorems used to find fixed points, so I believe the next step would be to investigate the analytic properties of this noncontinuous function.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Solve for positive real x

Without using fixed-point theorems, I can at least verify Mathsy’s assertion (but note that x = 0 is not a solution; x must be positive).

First, note that

for all integers

. This is because for all integers

,

(easily proved).

It remains to show that

for all integers

. This means showing that for all integers

or, since

for all integers

,

i.e.

Well, unless I’ve made a mistake somewhere, it is easy to see that the last line is true for all integers

,

Hence all

, where n is any integer greater than or equal to 2, are solutions.

Unfortunately we still need to show that there are no other solutions.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Solve for positive real x

Actually, we have that all

, where n is any positive integer, are solutions

.

So we have

and

.

Now suppose

, where

, is a solution.

Then we have

Then, in order for the solution to hold, we would need to have

This would mean

Now

and

.

If

, then

.

If

, then

.

In either case, we have a clear contradiction of

.

So this proves that there are no integer solutions other than

. (It is clear that all solutions must be integers since the LHS of the given equation is an integer.)

Last edited by JaneFairfax (2008-11-09 08:49:47)