Non-negative integer solution of the exponential Diophantine equation 2x+(pq)=z2
Letp,q are odd primes and p＜q.The non-negative integer solution of the Diophantine equation 2x-(pq)y=z2 was researched with the primary methods of congruence and quadratic remainder.It was proven that if(p,q)≡(1,3),(1,5),(3,1),(3,5),(3,7),(5,1),(5,3),(5,7),(7,3),(7,5)(mod8),then the Diophantine equation 2x-(pq)y=z2 has only non-negative integer solution(x,y,z)=(3,0,3),with the exceptions that the equation has only non-negative integer solutions(x,y,z)=(3,0,3),(0,1,4),(6,2,17)whenp=3,q=5 and the equation has only non-negative integer solutions(x,y,z)=(3,0,3),(0,1,p+1)whenq=p+2,p≠3.
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