# Roth's theorem and exponential diophantine equations

Can anyone give me some papers that show how the Thue-Siegel-Roth theorem can be applied to the theory of exponential diophantine equations? Or even an example of this application (any exponential equation would suffice).

• I believe Roth's theorem was a result motivated by wanting to solve equations of Thue type, whereas its generalization, the Schmidt subspace theorem, is the one with strong applications to exponential Diophantine equations (for example it implies that the S-unit equation has only finitely many solutions). Jun 30, 2013 at 0:22
• I was looking for more specific examples that dont require as much machinery to prove. Jun 30, 2013 at 0:40

The main application of the (generalized) Thue-Siegel-Roth theorem to exponential Diophantine equations comes from Schmidt's subspace theorem, and an extension to $p$-adic valuations due to H.P. Schlickewei, see Gjergij's comment.
An interesting exponential Diophantine equation is $$a^{x_1}-a^{x_2}=b^{y_1}-b^{y_2}.$$ For fixed integers $x_1>x_2>0$, $y_1>y_2>0$, $y_1>x_1$, $x_1,x_2$ coprime and $y_1,y_2$ coprime, one knows that this equation has only finitely many positive integer solutions $(a,b)$. The proof ultimately relies on Schmidt's subspace theorem. (The ABC conjecture implies that this Diophantine equation has only finitely many positive integer solutions $(a, b, x_1 , x_2 , y_1 , y_2 )$ with $a > 1, b > 1$, $x_1\neq x_2$ and $a^{x_1}\neq b^{y_1}$.)
Remark: An interesting application is also Waring's problem, where effective Roth-type inequalities for irrational algebraic numbers can be applied. Waring's problem is as follows: for $k \ge 2$ define $g(k)$ as the smallest positive integer $g$ such that any integer is the sum of $g$ elements of the form $x^k$ with non-negative integers $x_i$, i.e., for each positive integer $n$ the equation $$n=x_1^k+x_2^k+\cdots + x_m^k$$ has a solution if $m = g(k)$, while there is a $n$ which is not the sum of $g(k) − 1$ such $k$-th powers. K. Mahler proved that $g(k)=I(k)=2^k +[(3/2)k]−2$ for sufficiently large $k$, using a $p$-adic version of the Thue-Siegel-Roth Theorem (which is not effective, though, so that we do not know what sufficiently large means). For small $k$ this is already true, e.g., for $3 ≤ k ≤ 471 600 000$. But of course, there is still a gap (to sufficiently large).
• Is the displayed Diophantine equation correct as written? Three $a$'s and one $b$ look unusual. Jun 30, 2013 at 20:16