@**techreport**{RISC2331,author = {Rahul Athale},

title = {{Symbolic computation in number theory}},

language = {English},

abstract = {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.},

number = {04-20},

year = {2004},

length = {93},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}