One of the first systems developed to provide online security is what is now known as RSA encryption. This method has been first published by Rivest, Shamir and Adleman in 1978. Even on this day, a modification of the first method of RSA encryption is being used and considered safe. This paper explores a number of mathematical structures and describes whether or not it could be possible to extend RSA encryption. In this paper we will focus on verifying whether the RSA encryption and decryption algorithms are well defined on the mathematical structures.

The four-colour conjecture (4CC) is a question that asks whether any map can be coloured using only four colours, with the constraint that neighboring countries must have distinct colours. This conjecture has re- mained unanswered for over 170 years, with a rich history of various attempts to prove it. In 1879, Kempe put forward a proof, but it was invalidated by Heawood after 11 years. Heawood did succeed, however, in prov- ing the weaker five-colour theorem. It wasn’t until 1976 that the first genuine proof was discovered by Appel and Haken. This proof sparked controversy due to its reliance on approximately 1200 hours of computer computation, making it unverifiable by hand.
The purpose of this report is to provide a comprehensive overview of the historical background and re- cent advancements in the four-colour theorem. It will delve into the numerous failed attempts to prove the conjecture and discuss the groundbreaking proof by Appel and Haken. Additionally, it will explore a recent endeavor by Dr. Xiang, who approached the problem from a different perspective but also encountered a fallacy in his work. The main contribution of this thesis involves a new proof that builds upon the research of Dr. Yeh, who attempted to prove the four-colour theorem by transforming it into a system of linear equa- tions. The missing crucial steps will be addressed for whose correction new ideas will be introduced, using an integral version of Farkas’ lemma and superadditivity. The ideas being researched do not finalize the proof but contribute towards a possible elegant proof in the future. Finally, related topics such as maps on different surfaces and maps with disconnected regions will be considered, broadening the scope of the discussion.

Dit onderzoek gaat over de mogelijkheden en onmogelijkheden met betrekking tot constructies met passer en ongemarkeerde liniaal. Hier werken we met de euclidische meetkunde en zullen we dus ook beginnen met de vijf postulaten van Euclides en een aantal toepassingen ervan. Er zal besproken worden wat het betekent voor een punt of constructie om construeerbaar te zijn, hiervoor zal gedefinieerd moeten worden wat het betekent voor een getal om construeerbaar te zijn. Vervolgens komen de volgende meetkundige problemen aan bod: de driedeling van een hoek, de verdubbeling van een kubus, de kwadratuur van een cirkel en de construeerbaarheid van regelmatige veelhoeken. Om te laten zien wat wel en niet mogelijk is wordt er gebruik gemaakt van moderne algebra, zo kan een constructie probleem omgeschreven worden naar een concreet algebraïsch probleem. Hiervoor maken we gebruik van de theorie rondom lichaamsuitbreidingen en de Galois theorie.

Nowadays, we produce an enormous amount of data and all this data needs to be stored somewhere. In order to store all this data in an efficient and environmentally friendly way, solutions need to be found. A storage technique that has great potential to be a solution is synthetic DNA based storage.
A DNA-strand is made up of a sequence of four nucleotides, Adenine (A), Thymine (T), Guanine (G) and Cytosine (C). To save our data using DNA, our binary data is encoded to quaternary data, using this quaternary data DNA strands are made.
During this process, of saving our data in DNA, errors can occur, therefore two constraints are considered to minimise the number of errors in our codes.
The GC-weight, which is the number of G and C nucleotides in a DNA word and the maximal run-length, which is the maximum number of identical nucleotides in a row. In addition, the Hamming distance between words is also considered to be able to detect and even correct errors that still occur.

This research tries to find lower and upper bounds for the maximum size of DNA codes.
We continue the work of Van Leeuwen [1], Vermeer [2] and Laseur [3], who considered codes with fixed GC-weight. In this thesis we will relax this GC-weight constraint and we will look at codes where we have a set of GC-weights and all the DNA words in the code need to satisfy one of these weights. Furthermore, the codes we consider have a run-length of r=1. We will determine the maximum size of codes where the minimum Hamming distance d is equal to the length n of the DNA words. We determine lower and upper bounds for the case where d=n-1 and for specific sets of GC-weights we will determine the maximum size. Finally we will present nine algorithms, these algorithms create DNA-codes. The size of these codes give us lower bounds for the maximum size of DNA codes.