For any fixed odd prime p, let N（p） denote the number of positive integer solutions （x, y） of the equation y^2 = px（x^2 ＋ 2）. In this paper, using some properties of binary quartic Diophantine equations, we pro...For any fixed odd prime p, let N（p） denote the number of positive integer solutions （x, y） of the equation y^2 = px（x^2 ＋ 2）. In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7（mod 8）, then N（p） = 0; ifp ≡ 1（mod 8）, then N（p） 〈 1; if p〉 3 andp ≡ 3（rood 8）, then N（p） ≤ 2.展开更多
In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational...In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don＇t.展开更多
基金Foundation item： Supported by the Natural Science Foundation of Shaanxi Province（2009JM1006）
文摘For any fixed odd prime p, let N（p） denote the number of positive integer solutions （x, y） of the equation y^2 = px（x^2 ＋ 2）. In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7（mod 8）, then N（p） = 0; ifp ≡ 1（mod 8）, then N（p） 〈 1; if p〉 3 andp ≡ 3（rood 8）, then N（p） ≤ 2.
基金Supported by NNSF(No. 19931020, No. 10001006 and No. 60273015)of China
文摘In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don＇t.
