| The Diophantine equation not only develop activily, but also be apply to Discrete Mathematics else field. For example in the theory of numbers, the algebra, the combinatorics and so on the discrete mathematics discusses each kind of limited structure mathematics branch, many need to solve has the quite difficult question solution to have the possibility to sum up to certain indefinite equation solution. So it plays a key role in people's study and research and solving the actual problems.Several authors have studied on the Diophantne equation x3±23n =3Dy2.For D =/ 6 k+1,all positive integer solution of the two equations have been obtained. When D has not square factor, has not 3 factor, and has 6 k +1 prime factor, it has difficulty. For n =2, the equation is x3±64 =3Dy2. The equation x3±64 =3Dy2(0< 3 D<100),when D =7,13,19,31, has not been solution.In this paper, with the method of recurrence sequences , congruence, only decomposition, Maple formality, quadratic residue, and Pell equation,we have shown four Diophantine equations the only solution in positive integers, and we have discussed the Diophantine equation x3±23n =3Dy2simply ordinary integer solution sufficient conditions. This article is divided four chapters to show:In the first chapter, we summarize the present research situation home and abroad of the Diophantine equation x3±23n =3Dy2.In the second chapter, we give the prior knowledge to the whole paper and give a brief introduction to the characters of Pell equations,the origin of recurrence sequence and the method of congruence. In the third chapter, we prove the whole solutions of Diophantne equation x3±64 =3Dy2( D =7,13,19,31 ) in four sections; at the same time, section V of Diophantne equation x3±23n =3Dy2 simply ordinary integer solution to a sufficient condition for the continuation of discussions. In the fourth chapter, we summarize the total paper and put forward some problems which should be solved in the future.In this paper, main result gather in the third chapter. |
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