When adding coprime numbers A and B, one could ask how big A, B, and A + B could be compared with the product of the prime numbers dividing these numbers. One can expect that this prime product has...Show moreWhen adding coprime numbers A and B, one could ask how big A, B, and A + B could be compared with the product of the prime numbers dividing these numbers. One can expect that this prime product has about three times as many digits as A + B, but with smart choices of A and B this prime product can be smaller than A + B. However, the so-called ABC-conjecture says that it cannot be much smaller. Several mathematicians have tried to develop algorithms creating infinitely many triples An, Bn, and An + Bn such that An + Bn is large compared to the product of the primes dividing one of the numbers An, Bn, and An + Bn. And I add a new algorithm to this list of algorithms and use the tool of elliptic curves, the zero set of a polynomial equation together with a group law, to create my triples.Show less
In 1985 is het ABC-vermoeden bedacht door de wiskundigen David Masser en Joseph Oesterl´e. Het vermoeden is een vrij eenvoudig probleem, waar geen diepe wiskunde voor nodig is om het te begrijpen....Show moreIn 1985 is het ABC-vermoeden bedacht door de wiskundigen David Masser en Joseph Oesterl´e. Het vermoeden is een vrij eenvoudig probleem, waar geen diepe wiskunde voor nodig is om het te begrijpen. Het vermoeden bewijzen echter is een heel ander verhaal. Daar gaan wij ons in deze scriptie dan ook niet aan wagen. Voordat we verder kunnen gaan, moeten we eerst enkele begrippen introduceren.Show less
In my thesis I will generalize a previous result of M. Maehara. In “Distances in a rigid unit-distance graph in the plane”1 he proved that the distances that occur between vertices in planar rigid...Show moreIn my thesis I will generalize a previous result of M. Maehara. In “Distances in a rigid unit-distance graph in the plane”1 he proved that the distances that occur between vertices in planar rigid unit-distance graphs are precisely the positive real algebraic numbers. A unit-distance graph is a framework of equal length bars which are connected in a flexible way at their endpoints. Such a framework is called rigid if it cannot be deformed without changing the length of the bars. There is also a stronger notion of rigidity which is called infinitesimal rigidity. This stronger notion asks that the framework cannot even be deformed infinitesimally without an infinitesimal change of the lengths of the bars. The picture on this page shows an example of a framework which is rigid but not infinitesimally rigid. The point with the arrows, cannot really move relative to the rest of the construction, although you can do this infinitesimally in the direction indicated by the arrows. A yet unanswered question was whether Maehara’s result also holds for infinitesimally rigid frameworks. It turned out to be true even with this stronger notion of rigidity. I will prove this in my thesis by showing that Maehara’s construction is infinitesimally rigid in most cases and give a different construction for the cases where Maehara’s construction isn’t infinitesimally rigid.Show less
