A Study Of Several Problems In Number Theory

In this thesis,we study three relatively independent topics in modern mathemat-ics related to algebraic numer theory.They are:the construction of non-congruent numbers,the 2-adic properties of the number of domino tilings on a torus,and some congruences concerning quadratic ideal class numbers with continued fractions.This work is divided into three chapters.Each topic is discussed in one chapter.First of all,based on Monsky's formula for the 2-Selmer rank of congruent elliptic curves[20],it is possible to construct non-congruent numbers though the computation of block matrices over F2.In Chapter one,we construct some new families of non-congruent numbers.These new non-congruent numbers have arbitrarily many prime divisors,and their prime divisors may come from different residue classes modulo 8.Comparing with the known results(such as the results given by Keqin Feng and his students since 1990s based on elliptic curves and algebrai graph theory,12-15],and the recent major breakthrough on the famous congruent number problem and BSD conjecture given by Ye Tian and his co-participants[58]),the method used in this paper is more elementary but effective when confined to the non-congruent numbers.As we know,Lindsey Reinholz,Blair K.Spearman and Qiduan Yang used this method to find new non-congruent numbers even since 2013[48-51].Secondly,based on the notable results on the number of domino tilings of the finite quadratic lattice given by P.W.Kasteleyn in 1961[28],Henry Cohn studied the 2-adic properties of the number of domino tilings on a 2n × 2n square[8].In Chapter two,we further study the 2-adic behavior of the number of domino tilings on a(4n + 2)x(4n + 2)torus.Specifically,we show that this number can be written as 24n+2g(n)2 + 28n+2(2n + 1)4nh(n),where g(n)and h(n)are odd positive integers.We prove that g(n)and h(n)are uniformly continuous under the 2-adic metric as n varies and satisfy the functional equations g(-1-n)= g(n)and h(-1-n)= h(n)respectively.Thus we get an analog of Henry Cohn's result on a square.Finally,let h(-p)and h(p)be the class numbers of the quadratic number fields Q(?)and Q(?)respectively.Based on the class formulas followed from the work of Don Zagier on Kronecker limit formula in 1970s[63,64],in 2015,Lynn Chua,Benjamin Gunby,Soohyun Park and Allen Yuan proved that if p ? 3(mod 4)is a prime then the congruence h(-p)? h(p)m(p)(mod 24)holds,where m(p)is an integer deter-mined by the minimal period of the negative regular continued fraction expansion of(?)[5].In Chapter three,based on Zagier's above work and a new class for-mula given by Hongwen Lu which concerned quadratic class numbers with the Hirze-bruch sums.We further prove that if p ? 1(mod 8)is a prime then the congruence h(-p)? h(p)m(4p)(mod 8)holds;and if p ? 5(mod 8)is a prime,then the congru-ence h(-p)? h(p)m(4p)(mod 8)holds under certain conditions;where m(4p)is an integer determined by the minimal period of the negative regular continued fraction expansion of(?).