Wikipedia:Reference desk/Archives/Mathematics/2021 November 5

= November 5 =

Pillai's theorem: $$|Ax^n - By^m| \gg x^{\lambda n}$$
For every $$x>1,y>1,n>1,m>1$$, Pillai's theorem states the following:
 * The difference $$|Ax^n - By^m| \gg x^{\lambda n}$$ for any λ less than 1, uniformly in m and n.

I wonder if one can prove (e.g. by his theorem) the following claim:
 * There exists $$x$$, such that for every $$n>2, m>1$$, there exist $$a,b$$ such that every $$A>a,B>b$$ satisfy $$|Ax^n - By^m| > (x+2)^{2}$$.

If the answer is positive, and $$P_x$$ is the minimal prime larger than any given $$x$$, then I also wonder if one can prove (e.g. by his theorem) the following:
 * There exists $$x$$, such that for every $$n>2, m>1$$, there exist $$a,b$$ such that every $$A>a,B>b$$ satisfy $$|Ax^n - By^m| > P_x^{2}$$.

185.24.76.181 (talk) 14:11, 5 November 2021 (UTC)


 * I'd like to see a precise statement of the theorem, written out with explicit quantifiers. Perhaps I misunderstand the $${\gg}$$ notation, but I doubt that, uniformly,
 * $$|(x^2)^n - x^{2n}| \gg \sqrt{x}.$$
 * --Lambiam 18:31, 5 November 2021 (UTC)
 * Well, he at least means that for every $$x,y$$ and every $$n>1, m>1$$, there exist $$a,b$$ such that every $$A>a,B>b$$ satisfy $$|Ax^n - By^m| > x^{\lambda n}$$ for any λ less than 1. 185.24.76.176 (talk) 19:23, 6 November 2021 (UTC)
 * So set $$x=y=1,n=m=2,$$ and given $$a$$ and $$b$$ with the stated property whose existence is promised for these values, set $$A=B=\operatorname{max}(a,b)+1.$$ Then the lhs of the inequation equals $$0,$$ while the rhs equals $$1.$$ --Lambiam 22:57, 6 November 2021 (UTC)
 * Well, I was wrong with my interpretation. Reading our article about Pillai's theorem, I'm sure he at least meant that for every $$x>1,y>1,n>1,m>1$$, there exist $$a,b$$ such that every $$A>a,B>b$$ satisfy $$|Ax^n - By^m| > x^{\lambda n}$$ for any λ less than 1. 185.24.76.176 (talk) 23:26, 6 November 2021 (UTC)
 * Then set $$x=y=2,$$ and the rest as before. --Lambiam 23:55, 6 November 2021 (UTC)
 * Oh, so weird! Thanks to your comment, now I wonder what our article means - quoting Pillai's theorem. 185.24.76.176 (talk) 10:53, 7 November 2021 (UTC)