【摘要】 研究了一类丢番图方程ｘ^３±(２^２ｋ－１)^３＝Ｄｙ^２．给出了无非平凡整数解的充分性条件以及非平凡整数解的个数与求法。更多还原

【Abstract】 It have intended,among others to prove the following:Theorem 1. If odd D>2,and d|D,d is not a square,d≠3,and d is not prime of the from 6m+1 ,then equation x￣3+ (2￣(2k-1))￣3 =Dy￣2 ,k≥3.1) when D=4l+3, except Da￣2=3(2￣(4k-6)+2￣(2k-2))-1, x=3·2￣(4k-6)+2￣(2k-2)-1 ,b = 3·2￣(4k-6)+1,y=ab, maybe produce one solution (D,x,y),the equation have not 2x nontrivial integer solution. Except (D_i,4x_i,8y_i),i=1,2,…,t,2≤t≤k-1,the equation have not 2|x nontrivial integer solution,where (D_i ,x_i,y_i) ,i=1, 2,…,t are all nontrivial integer solutions of the equation x￣3 + (2￣(2(k-1)-1))￣3 =Dy￣2.2)when D= 4l+1,except Da￣2= 2￣(4k-6)+ 3·2￣(2k-2)-3,x=-2￣(4k-6)+2￣(2k-2)-3,b=2￣(4k-6)+3,y=ab maybe produce one solution (D,x,y) ,the equation have not 2x nontrivial integer solution. Theorem 2. If odd D>2,and d|D,d is not a square,d≠3,and d is not prime of the from 6m+1, then equation x￣3-(2￣(2k-1))￣3=Dy￣2, k≥3.1) when D=4l+1,except Da￣2=2￣(4k-6)-3·2￣(2k-2)-3, x=2￣(4k-6)-2￣(2k-2)- 3,b=2￣(4k-6)+3,y=ab, maybe produce one solution (D,x ,y), the equation have not 2x nontrivial integer solution. Except(D_i,4x_i,8y_i). i=1,2,…,t,2≤t≤k-1,the equation have not 2|x nontrivial integer solution ,where (D_i,x_i ,y_i),i=1,2,…,t are all nontrivial integer solutions of the equation x￣3-(2￣2(k-1)-1)￣3 =Dy￣2. 2) when D=4l+3,except Da￣2=3(2￣(4k-6)-2￣(2k-2))-1,x= 3·2￣(4k-6)-2￣(2k-2)-1, b=3·2￣(4k-6)+1,y=ab maybe produce one solution (D,x,y),the equation have not 2x nontrivial integer solution.更多还原