In a message dated 03/04/03 02:34:44 GMT Daylight Time,

jack@...
writes:

> > Is it true...

> > that there exists x,y, a,b>1 s.t. x^a - y^b = n, for all n integers?

>

> Probably not. I don't think any solution is known for n=6, and I

> suspect that no solution exists for n=6. For n=6, I think that it

> can be shown easily that if a solution exists, one of the exponents

> must be >= 5, and the heuristics (and the scarcity of 5th-and-above

> powers) would lead one to strongly suspect that no solution exists.

>

I agree that heuristics based on density of powers would indicate "probably

not".

Since only 0 and 1 are quadratic residues mod 4, a = b = 2 is no good for n =

2 mod 4.

The next best bet, in terms of density of powers, is a=2, b=3. A couple of

months ago, as a by-product of investigating the "Fermat-Catalan conjecture",

I established that for (x^2) and (y^3) both <= 10^10, no less than 736754 of

the numbers <= 1 million could /not/ be represented as abs(x^2-y^3), the list

starting as follows:-

6,14,21,29,32,34,42,46,51,58,59,62,66,69,70,75,77,78,84,85,86,88,90,93,96,...

(Any pattern there, I wonder?)

Mike Oakes

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