Diophantine equations, arithmetic progressions and their applications

The main part of this project is devoted to Diophantine equations, with special interest in applications in various areas of number theory. In this project we are mainly interested in two major topics (additive unit structure of rings and arithmetic progressions on curves) and one minor topic (discrepancy theory). The first topic is the the so-called unit sum number problem for number fields. This problem is to characterize all number fields K such that the maximal order of K is generated by its units. Recently this topic gained interest by the investigations due to Ashrafi, Hajdu, Jarden, Narkiewicz and Vmos. Several instances of the problem have been solved by the proposer in cooperation with several coauthors (pure cubic, pure complex quartic and bi- quadratic cases). Beside the classical unit sum number problem we are also interested in several variations of this problem. E.g. the existence of power integral bases consisting of units, the generation of orders by distinct units and also analytic aspects of the problem are of interest within this project. In order to successfully attack these problems beside symbolic computation techniques also the theory of Diophantine equations will be a key point. Results concerning binary Thue-equations, simultaneous Pell-equations, index-form equations and many more types of Diophantine equations play a key roll in this part of the project. The second large topic within this project are arithmetic progressions on (plane, algebraic) curves. Given an equation f(x,y)=0 for a plane, algebraic curve we say that this curve admits an arithmetic progression of length n if there exist n rational points on the curve such that the x-components (or the y-components) form an arithmetic progression. In this project we are mainly interested in quadratics (Pell equations), elliptic curves and hyper-elliptic curves. In the case of elliptic curves this problem goes back to Mohanty who showed that there do not exist 5 consecutive integers that are the y-component of an elliptic curve of the form y2 =x 3 +k with an integer k>0. Further Mohanty conjectured that there are no arithmetic progressions of length 4 on such curves. More generally we want to study the numbers m(d) and M(d) respectively, which are the largest integer n such that there is one respectively infinitely many polynomials g of degree d such that the curve y2 =g(x) admits arithmetic progression of length n. Such questions are also related to the magic square of squares problem. Another application of Diophantine equations has been found in the context of discrepancy theory and metric number theory. First, by the use of the Subspace-theorem due to Schmidt one can show that certain low discrepancy sequences avoid corners, which has implications on the Quasi-Monte-Carlo integration of unbounded integrands. On the other hand the property of a sub-sequence of an n-alpha sequence to be a low discrepancy sequence is closely related to the number of solutions to certain S-unit equations. Within this project we also plan to consider these aspects.

In this project three major results were established. Firstly an old problem dating back to Diophantus of Alexandria from ancient times was solved, the so-called quintuple conjecture. Secondly a problem of Jarden and Narkiewicz on the additive structure of number fields was solved. And thirdly a question concerning the prime divisors of the sum of two Fibonacci numbers was studied. Diophantus of Alexandria (around 300 AD, exact biographic dates do not exist) asked for four numbers such that taking two of them multiplying together and adding one gives a perfect square. A quadruple with this property is called nowadays a Diophantine quadruple in honor of Diophantus. For example Fermat found the Diophantine quadruple 1, 3, 8, 120. This topic was also picked up by Euler, who showed that there exist infinitely many Diophantine quadruples. However it remained an open problem whether five positive integers with this property exist, i.e. whether a Diophantine quintuple exists. Within this project this question was treated and we found a proof that no Diophantine quintuple exists and therefore settling a long outstanding question. The second major result concerns the additive structure of number fields. Given a number field K we ask which proportion of its algebraic integers can be written as a sum of k units, where k is a fixed number. This question was first asked by Jarden and Narkiewicz in 2007 and was later solved subsequently by Filipin, Frei, Fuchs, Tichy and Ziegler in a slightly modified version. However the original question how many positive integers n = x can be written as a sum of k units remained unsolved. Within this project it was possible to answer the question of Jarden and Narkiewicz in its original form. The Fibonacci sequence (fn )8 n=0 is defined by f0 = f1 = 1 and fn+2 = fn+1 + fn for n = 2. Many mathematicians have investigated this sequence going back to Fibonacci who was the first to study this sequence in 1202. In this project we determined all pairs of Fibonacci number (fn , fm ) with n > m such that their sum fn + fm is divisible only by prime numbers p = 200. In particular, we showed that for such pairs we always have that n < 63.