Common solution of Pell equations x2-40y2=1 and y2-Dz2=9
Let D=2p1…ps(1≤s≤4),where p1,…,ps are distinct odd prime numbers.The common solutions of Pell equations x2-40y2=1 and y2-Dz2=9 are studied by using parity analysis,congruence,recursive sequence and properties of the solutions of Pell equations.It is concluded that when D ≠ 2×7×103,the system have only trivial solution(x,y,z)=(±19,±3,0);when D=2×7×103,in addition to the trivial solution(x,y,z)=(±19,±3,0),there is non-trivial solution(x,y,z)=(±27 379,±4 329,±114).The results enrich the research content of the integer solution of this kind of Pell equations.
Pell equationodd and even analysisodd prime numbercongruence
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