Find the total number of integral solutions to the equation
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Solving for y in terms of x gives y = x 2 − 7 2 x 2 = 1 + x 2 − 7 2 7 2 . We are interested in those x ∈ Z such that ( x 2 − 7 2 ) ∣ 7 2 , and the integer divisors of 72 include:
± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 8 , ± 9 , ± 1 2 , ± 1 8 , ± 2 4 , ± 3 6 , ± 7 2 .
Testing each of these divisors gives integral values for x , y for:
7 2 ⇒ x = ± 1 2 , y = 2 ;
9 ⇒ x = ± 9 , y = 9 ;
− 8 ⇒ x = ± 8 , y = 8 ;
− 3 6 ⇒ x = ± 6 , y = − 1 ;
− 7 2 ⇒ x = 0 , y = 0 .
Thus, there are a total of 9 integral pairs that solve our original equation.