Research On The Integer Solutions Of Two Types Of Diophantine Equations

In this paper,by applying some elementary number theory me-thods such as recurrent sequence,the properties of the solutio-ns to Pell equation and congruence,we can obtain the following results:1.On the system of Diophantine equations X2-26y2=1 and y2-Dz2=100,we proved the following conclusions:(i)Let D=2p1…Ps,1?s?4(p1,…,ps are distinct odd primes),then it has only trivial solutions(x,y,z)=(±51,±10,0)with the exception that D=2×7×743;it has integer solutions(x,y,z)=(±530451,±104030,±1020),(x,y,z)=(±51,±10,0)where.D=2×7×743.(ii)Let D=2n(n?N),then it has only trivial solutions(x,y,z)=(±51,±10,0).2.Let r=st(s,t?N)and s is squarefree,A(r)=s,B(r)=t.On the Diophantine equation x3-1=709qy2,we obtained the following conclusions:Let a be an odd prime with q?1(modl2),we show a necessary and sufficient condition q=A(3×709 a4+2127a2+1),a?N for the Diophantine equation x3-1=709qy2 to have solutions,the so-lution is(x,y)=(1+2127a2,3aB(3×7092a4+2127a2+1)).In addition,let q satisfy one of the four conditions:q=12k2+12k+1(k?N),q=108k2±12k+1(k?N),q=12k2(+1 q ?1(mod 12)is an odd prime and(q/709)=-1,then the equation x3-1=709qy2 has no positive integer solution.