Diophantine方程组x-1=3pqa^2和x^2+x+1=3b^2

The System of Diophantine Equations x-1=3pqa^2 and x^2+x+1=3b^2

设p,q是适合3<p<q的奇素数,根据二次和四次Diophantine方程的结果,运用初等数论方法证明了:仅当(p,q)=(7,181)时方程组x-1=3pqa2和x2+x+1=3b2有正整数解(x,a,b)=(60 817,4,35 113).

Let p,q be odd primes with 3〈p〈q.By using the results of quadratic and quartic Diophantine equations with some elementary number theory methods,it is proved that the system of equations x-1=3pqa2 and x2＋x＋1=3b2 has only the positive integer solution（p,q,x,a,b）=（7,181,60 817,4,35 113）when（p,q）=（7,181）.