The Pell Equations x^2-8y^2=1 and y^2-Dz^2=1

摘要

In this paper,we have proved that if one of the following conditions is satisfed,then the equations in title has no positive integer solution:①D=∏si=1P i or D=2∏si=1P i and { P i≡3 (mod 4)} (1≤i≤s) or P i≡5 (mod 8) (i≤i≤s); ② D=∏si=1P i-1 (mod 12), 1≤s≤7 and {D≠3·5·7·11·17·577,7·19·29·41·59·577;} ③ D=2∏si=1P i,1≤s≤6 and {D ≠2·17,2·3·5·7·11·17,2·17·113·239·337·577·665857;} ④ D=∏si=1P i≡-1 (mod 12), 1≤s≤3 and D≠ 5·7,29·41·239.