||We used symbolic computation methods to analyse two number theoryproblems. We implemented some of these methods in the computer algebrasystems Mathematica, Maple, and Macaulay. So the thesis consists oftwo parts. The first part deals with the work on prime gaps and the secondone is about the generation of elliptic curves with high rank.We carried out extensive computations to determine the validity of theconjecture regarding takeover point of 210 as the most frequent prime gapfrom 30. Also, we wrote a program in Mathematica to compute the approximatenumber of gaps up to a given positive integer. We apply statisticaltests to the computed data and based on the results of those tests, we improvethe takeover point in the jumping champion conjecture. We alsoconsider the prime gaps modulo 6. We formulate a new conjecture basedon the following observation: The number of gaps congruent to 0 modulo6 equals approximately the number of gaps not congruent to 0 modulo 6.In the second part, we discuss the method suggested by Yamagishi forthe generation of the elliptic curves with high rank. We studied this approachextensively and implemented the method in Maple. We foundsome examples where this method does not produce the elliptic curveswith desired rank. We suggest certain constraints on the parameters in Yamagishi’smethod to get the elliptic curves with desired rank in the case ofrank 2. We also prove one of the required results using Macaulay.