Congruent Number and Related Topics

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A positive integer n is called a congruent number if it is equal to the area of a right triangle with rational sides. Determining whether a given positive number is congruent or not is known as the congruent number problem. It is well-known that n is a congruent number if and only if the rank of the Mordell-Weil group consisting of all rational points on the congruent number elliptic curve En : y2 = x3􀀀n2x is positive. Although congruent numbers have been studied for centuries, the problem of providing a complete classification still remains elusive. Various mathematicians constructed infinite families of congruent and non-congruent numbers with prime factors that satisfy certain congruence conditions. We begin the thesis by introducing congruent number and its generalization in chapter 1. In chapter 2, we briefly mention certain preliminaries from basic algebra, number theory and elliptic curves that we need later. Then we outline the method of complete 2-descent which plays a central role in our work. We conclude the second chapter with a description of Monsky’s matrices that we use subsequently. In chapter 3, we construct infinite families non-congruent numbers with arbitrarily many pairs of prime factors generalizing results of Lagrange [36] and Serf [46]. We use the method of complete 2-descent adopted earlier by Iskra [28] for constructing non-congruent numbers with prime factors congruent to 3 modulo 8. In chapter 4, we construct families of highly composite non-congruent numbers by considering Monsky’s matrices introduced in the appendix of [26]. The notion of -congruent number is a generalization of congruent number, where one considers the area of a triangle with all possible angles such that cos is rational rather than just = 2 (see [19], [31]). In chapter 5 we prove a criterion for a natural number to be a -congruent number over certain classes of real number fields. Tunnell [54] and Kazalicki [32] investigated the 2-part of class number of an quadratic imaginary field Q(p 􀀀p) where p is congruent prime number equivalent to 1 modulo 8 by studying congruence between certain half-integral weight modular forms. In chapter 6, we prove a divisibility result for the class number of Q(p 􀀀pq), where p and q are distinct primes satisfying (p; q) (5; 7) (mod 8) and pq is a congruent number. Rather than modular forms of half-integral weight, we exploit the method of complete 2-descent. Steuding [50] and Komatsu [35] considered the continued fraction expansion of some special types of irrational numbers (such as pn2 + 1 or p n2 + 2), whose limit is related to rational right triangles of area close to certain natural number. Keeping that perspective in mind, we have studied the period of the regular continued fraction of certain quadratic irrationals pn though we have not yet been able to link our findings to the question of n being congruent or not. In chapter 7, we include our results concerning the period of theregular continued fraction of p pq where p < q are two primes congruent to 3 modulo 4.We prove that the length of the period is divisible by 4 when q is a quadratic non-residue modulo p and is of the form 4k + 2 when q is a quadratic residue modulo p. We further examine the parity of the the central term in the palindromic part of the period of ppq. . We conclude the thesis by outlining scope of future research in chapter
Supervisor: Anupam Saikia