Study Of Several Special Kinds Of Diophantine Equations In Quadratic Fields

Diophantine equation, as a branch of number theory, has a long history and rich content. The indeterminate equation refers to the range of solutions of an integer, positive integer, rational or algebraic equation or integral equations, etc. Generally, the numbers of equation are much than unknown numbers of the equations. It is associated with algebraic, combinatorics, and the science of computer. So many scholars have a high interest in Diophantine equation.The main job of this thesis is to use algebraic number theory, the ideal of quadratic fields, class number and other relevant knowledge, to discuss the integer solutions of the Diophantine equation.First of all, we will be discussing the integer solutions of the Diophantine equation x2+D=4y5 (D=15,23,39,43,67,71). In this part, we will demonstrate that the equations x2+15=4y3,x2+23=4y5, x2+39=4y5, x2+67=4y5, x2+71=4y5 have not integer solutions.Secondly, we will be discussing the integer solutions of the Diophantine equation x2+C=yn (C=4,11,19,4n). And we will demonstrate that the integer solutions of the equationχ2+11= y3 are (χ,y)= (±4,3)(±58,15); the integer solutions of the equation x2+19= y3 is (x,y)= (±18,7),and the integer solutions of the equation x2+4n=y3 (χ=0( mod2))are(x,y,n)=(0,4k,3k),(±2×8k,±2×8k,3k+1)(±2×8k,±2×8k,3k+1),k∈N,k≥1;the equationx2+4= y5has no solutions.