关于不定方程4x^2-py^2=1

On the Indefinite Equation 4x^2-py^2=1

研究了二次不定方程4x2-py2=1(p为奇素数),对于特例p=m2±2(m为正奇数),利用Pell方程x2-py2=1的正整数解公式得到了原方程的所有正整数解.另外还证明了p≡1,5(mod8)时方程4x2-py2=1无正整数解.

In this paper,the quadratic indefinite equation 4x^2-py^2=1（p be an odd prime） is investigated.For special cases p=m2±2（m be a positive odd integer）,and we give the formulas of the positive integral solutions of the equation by using the Pell＇s equation x^2-py^2=1.Furthermore,we also prove that the equation 4x^2-py^2=1 has no positive integral solution when p≡1,5（mod 8）.