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#### Generalization of Ruderman's Problem to Imaginary Quadratic Fields

(2012-04-13)

In 1974, H. Ruderman posed the following question: If $(2^m-2^n)|(3^m-3^n)$, then does it follow that $(2^m-2^n)|(x^m-x^n)$ for every integer $x$? This problem is still open. However, in 2011, M. R. Murty and V. K. Murty ...

#### Arithmetic problems around the ABC conjecture and connections with logic

(2014-04-28)

The main theme in this thesis is the ABC conjecture. We prove some partial results towards it and we find new applications of this conjecture, mainly in the context of B\"uchi's n squares problem (which has consequences ...