On Several Diophantine Equations

This thesis mainly focuses on several Diophantine equations, and some results are shown as follows:1. In the first part, we study two Fermat's equations and mainly obtain the fol-lowing results:(1.1) Let p be an odd prime such that br + 1 = 2pt, where b, r, t are positive integers such that b = 3,5 (mod 8). We show that the Diophantine equation x2 +bm= pn has only the positive integer solution (x,m,n)=(pt-1,r,2t). We also prove that if b = q is a prime and r = t = 2, then the above equation has only one solution for the case q ? 3,5,7 (mod 8) and the case d is not an odd integer great than 1 if q ? 1 (mod 8), where d is the order of prime divisor of ideal (p) in the ideal class group of imaginary quadratic field Q((?)).(1.2) Let p ? q be two primes. Then the equation p2m - qn = z2 has at most one non-trivial solution (m, n, z) in natural number except q = 2. Moreover, there are only finite solutions (x, y, z, m, n) to the equation x2m - yn = z2 in natural number such that x > y be two consecutive primes.2. In the second part, we study the additive structure on multiplicative objects.And mainly obtain the following results:(1.1) Let K be an algebraic number field with OK its ring of integers, and n a nonzero ideal of OK. For an element a ? OK/n, we define (OK/n)*·a as an orbit of a. Then we show explicitly which orbits are part of the union which constitutes the sumset of two given orbits. We also obtain a formula for the number of representations of each element in the sumset of two orbits.(1.2) We give a formula for sums of exceptional units in residue class ring and obtain an identity.3. Let p, q be two different odd primes satisfying q - p = 2s, where s ? 1. We consider a family of elliptic curves E :y2 = x(x + 2tp)(x + 2tq),where t ? 0 is integer. In this part, we obtain some formulas for the rank of E(Q)and the dimension of Shafarevich-Tate group. Furthermore, if we assume the Birch-Swinnerton-Dyer Conjecture, then we can give the rank of E(Q) in some special cases.