Re: Difference of like powers of rational numbers

On Wed, 16 May 2007 23:54:30 -0500, Robert Israel
<israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

quasi <quasi@xxxxxxxx> writes:

On Thu, 17 May 2007 00:35:24 -0500, quasi <quasi@xxxxxxxx> wrote:

On Thu, 17 May 2007 00:25:08 -0500, quasi <quasi@xxxxxxxx> wrote:

On Wed, 16 May 2007 13:57:10 -0500, quasi <quasi@xxxxxxxx> wrote:

Conjecture 1:

If x,y are distinct nonzero real numbers such that

x^n-y^n is an integer
for all sufficiently large positive integers n

then x,y must both be integers.

Conjecture 1c:

The same as conjecture 1 except replace "real" by "complex".

Conjecture 1c has been proved (Robert Israel's proof and also my
proof), but I can envision stronger results.

How about this conjecture:

If x,y are nonzero complex numbers such that

x^n-y^n is an integer for infinitely many positive integers n

and (x/y)^m is not equal to 1 for any positive integer m

then x,y must both be integers.

Whoops -- I can see simple counterexamples.

For example, let x=2, y=i.

I'll have to rethink this.

quasi

In fact, counterexamples all over the place.

Here's another one: x=sqrt(3), y=sqrt(2).

Ok, I'll try one last conjecture for tonight ...

Conjecture:

If x,y are distinct nonzero complex numbers such that

x^n-y^n is an integer for infinitely many coprime positive integers n

then x,y must both be integers.

Counterexample...
x = sqrt(a)+b, y = sqrt(a)-b where a and b are nonzero integers and
sqrt(a) is not an integer. Then x^n - y^n is an integer for all odd
positive integers n.

Hehe.

That was quick.

Nice example.

Ok, here's my absolute last conjecture for tonight:

Conjecture:

Let x,y be complex numbers.

Let S be the set of positive integers n such that

x^n-y^n is rational

If S is infinite, then S contains an infinite arithmetic progression.

quasi
.

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