Computational experiences on norm form equations with solutions forming arithmetic progressions

In the present paper we solve the equation NK/Q(x0 + x1α + x2α2 + ... + xn -1αn -1) = 1 in x0, ... , xn -1 ∈ Z, such that x0, ... , xn -1 is an arithmetic progression, where α is a root of the polynomial xn - a, for all integers 2 ≤ a ≤ 100 and n ≥ 3

On the diophantine equation xp−x=yq−yx^p-x=y^q-y

We consider the diophantine equation x^p-x=y^q-y \tag"$(*)$" in integers $(x,p,y,q)$. We prove that for given $p$ and $q$ with 2\le p < q $(*)$ has only finitely many solutions. Assuming the abc-conjecture we can prove that $p$ and $q$ are bounded. In the special case $p=2$ and $y$ a prime power we are able to solve $(*)$ completely

A new push‐pull sample design for microscale mode 1 fracture toughness measurements under uniaxial tension

The miniaturization of microelectronic devices and the use of thin hard coatings have led to an increased demand for knowledge on the fracture behaviour of microscopic structures. A new geometry called Micro-SENT is proposed in this work that allows performing experiments in uniaxial tension on the microscale using a standard flat punch indenter by making use of a symmetric push-pull sample design. This enables the measurement of mode 1 fracture toughness under uniform tensional far-field loading as opposed to current state of the art approaches based on cantilever bending or micropillar splitting. Please click Additional Files below to see the full abstract