"Symbolic computation in number theory"
, Serie RISC Report Series, University of Linz, Austria, Nummer 04-20, RISC, Johannes Kepler University, 4040 Linz, Austria, 2004
Symbolic computation in number theory
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We used symbolic computation methods to analyse two number theory problems. We implemented some of these methods in the computer algebra systems Mathematica, Maple, and Macaulay. So the thesis consists of two parts. The first part deals with the work on prime gaps and the second one is about the generation of elliptic curves with high rank. We carried out extensive computations to determine the validity of the conjecture regarding takeover point of 210 as the most frequent prime gap from 30. Also, we wrote a program in Mathematica to compute the approximate number of gaps up to a given positive integer. We apply statistical tests to the computed data and based on the results of those tests, we improve the takeover point in the jumping champion conjecture. We also consider the prime gaps modulo 6. We formulate a new conjecture based on the following observation: The number of gaps congruent to 0 modulo 6 equals approximately the number of gaps not congruent to 0 modulo 6. In the second part, we discuss the method suggested by Yamagishi for the generation of the elliptic curves with high rank. We studied this approach extensively and implemented the method in Maple. We found some examples where this method does not produce the elliptic curves with desired rank. We suggest certain constraints on the parameters in Yamagishi's method to get the elliptic curves with desired rank in the case of rank 2. We also prove one of the required results using Macaulay.