Applying the Thue-Siegel Theorem Let $p(n)$ be the greatest prime divisor of $n$. Chowla proved here that $p(n^2+1) > C \ln \ln n $ for some $C$ and all $n > 1$. 
At the beginning of the paper, he mentions briefly that the weaker result $\lim_{n \to \infty} p(n^2+1) = \infty$ can be proved by means of the Thue-Siegel theorem (note: it was written before Roth considerably improved the theorem). 


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*Can someone elaborate on this? I was able to reduce the problem to showing that the negative Pell equation $x^2-Dy^2 = -1$ (where $D$ is positive and squarefree) has a finite number of solutions such that $p(y)$ is bounded by some $C$.

*Can someone provide examples of applications of Thue-Siegel in Diophantine equations? In the Wikipedia article it is mentioned that "Thue realised that an exponent less than d would have applications to the solution of Diophantine equations", but I am not sure I am realising it...

Suppose $p(n^2+1)$ was bounded by $X$, then no matter how large $n$ is $n^2+1$ factors into a product of powers of primes below $X$. In particular we have the factorization $n^2 + 1 = A^4 B^2 C$ where $B$ and $C$ are square free. There are $2^{|X|}$ possible values $C$ can take and $2^{|X|}$ $B$ can take. Some fixed $B^2C$ (with $C>1$) must occur for infinitely many $n$, in which case we have $$\frac{1}{A^4} = \left(B \sqrt{C}-\frac{n}{A^2}\right)\left(B \sqrt{C}+\frac{n}{A^2}\right)$$ infinitely many approximations of order 4 quality contradicting Thue-Siegel ($n$ may not be coprime to $A^2$, but by pigeonhole you can find some subsequence of $n$s for which they are all coprime to the same factor of $A^2$).
Edit Sorry this must be wrong since we can prove the same result by completely elementary methods. I'm not sure what the $\lim$ about $p$ means in this case.
Edit2: we need to show that the largest prime factor of n^2+1 is bounded below by an increasing function
just proving that n^2+1 has arbitrarily large prime factors isn't enough
it needs to be shown that it will always have large prime factors for n large
