The integer solutions of the cubic Diophantine equation x3±33=pqy2

Abstract

The solvability of a class of cubic Diophantine equations is studied by using properties of congruence, Legendre symbol and the methods of elementary number theory. The following several results are obtained: If p=3(24r+19)(24r+20)+1(r∈Z+) is odd prime, Diophantine equation x3±33=pqy2 has no positive integer solution；If both p≡13(mod24)and q=12s2+1(s∈Z+, 2 s) are odd primes, Legendre (p/q)=-1, Diophantine equation x3-33=pqy2 has only trivial solution (x, y)=(3, 0).

Keywords

• diophantine equation
• odd prime number
• legendre symbol
• integer solution