一类Pell方程组的非负整数解

Nonnegative Integer Solutions on a Sort of Pell Equations

设p,q 1,q 2为不同的奇素数,且p=2s+1,s≡2(mod 4),D=2c(c∈N,c=q_(1)或q_(1)q_(2)).证明了Pell方程组X^(2)-s(s+1)Y^(2)=1和Y^(2)-DZ^(2)=4(X,Y,Z∈N)除平凡解(X,Y,Z)=(2s+1,2,0)外,当2p^(2)-1=cr^(2)(r∈N)时,仅有解(X,Y,Z)=(4p^(3)-3p,8p^(2)-2,4pr).

Let p,q_(1),q_(2) be diverse odd primes,and p=2s+1,s≡2(mod 4),D=2c(c∈N*,c=q_(1) or q_(1)q_(2)).The following conclusion are proved:If 2p^(2)-1=cr^(2)(r∈N*),then Pell equations G:X^(2)-s(s+1)Y^(2)=1 and Y^(2)-DZ^(2)=4(X,Y,Z∈N)has only a solution(X,Y,Z)=(4p^(3)-3p,8p^(2)-2,4pr)with the exception that G has only trivial solution(X,Y,Z)=(2s+1,2,0).