On The Diophantine Equations Y~2=X~3+K(-100≤k≤100)

Diophantine equations are not only active in itself, but also are broadly applied to the fields in Discrete Mathematics. They play a key role in studying and solving actual problems. Therefore there are many scholars home and abroad who do broadly and deeply research in them.Many scholars have done lots of work on the Diophantine equations Ay~2 +B=xn. Especially for A =1and B =1, Lebesgue has proved that there are no solutions for the equations.And Rebinowitz got all the whole integer solutions for the Diophantine equations x p + 2 2 m=py2(x,y∈Z)when p =3.At the year 1842,Catalan guessed that :except the only one solution 3 2 - 23=1 ,there is on interger solutions for the Diophantine equations x~m - y~n=1( x, y,m,n∈Zx > 1, y >1,m>1,n>1),which was totally soluted by KeZhao at the year of 1982.But the Diophantine equations y~2 = x~3+kthat studied in this paper ,are not completely soluted.In this paper, with the methods of recurrence sequences, congruence, the characters of Pell equations, quadratic residue and the algebraic theory of numbers, we study the Diophantine equations y~2 = x~3+k(-100≤k≤100),got some results for k is square number and the cubicnumber.In the paper,there are four chapters. .In the first chapter, we summarize the present research situation home and abroad of the Diophantine equations y~2 = x~3+k(-100≤k≤100). 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 relevent knowledge about the algebraic theory of numbers and the method of congruence.In the third chapter, we discussed the equations y~2 = x~3-a~2,y~2 = x~3+9, y~2 = x~3+( pn)2, y~2 = x~3±8和y~2 = x2-(3m)2nseperately about the answers and ways. In section I, we prove that there is no solutions for the equation y~2 = x~3-a~2. In section II, we prove that the Diophantine equation y~2 = x~3+9 has only integer solution ( x, y)= (3,±6)and ( x, y)= (-2,±1). In section III, we give the methods of how to solute the Diophantine equation y~2 = x~3+( pn)2.And then goes the section IV, we prove that the Diophantine equation y~2 = x~3±8 can change to the integer solutions about the Pell equation n 0 2- 3m 02=1. In the last section V, we prove that the Diophantine equation y~2 = x2-(3m)2n has no interger solutions when y≡1(mod2),and m≡0(mod2).In the fourth chapter, we summarize the total paper and put forward some problems which should be solved in the future.