diophantine equation: $ (1-ab^3)(a^3b-1)=c^2$

Question

I wonder if the Diophantine equation $(1 -ab ^ 3 ) (a ^ 3b -1) = c^2$ admits rational solutions

we must choose the numbers $a$ and $b$

Answer 1

one infinite family is $$ b = \frac{1}{a} $$

There are other types of solutions. I took integers $A,B,D$ such that $$ (D^4 - A B^3 ) (A^3 B - D^4) = M^2 $$ are all integers. Then for your problem, take $$ a = A / D, \; \; \; b = B/D $$

   rat    a =  123 / 40     b =  2 / 3  INT  B: 80 =  2^4 5  D: 120 =  2^3 3 5 A: 369 =  3^2 41
   rat    a =  3 / 2     b =  40 / 123  INT  B: 80 =  2^4 5  D: 246 =  2 3 41 A: 369 =  3^2 41
   rat    a =  3 / 1     b =  193 / 291  INT  B: 193 =  193  D: 291 =  3 97 A: 873 =  3^2 97
   rat    a =  291 / 193     b =  1 / 3  INT  B: 193 =  193  D: 579 =  3 193 A: 873 =  3^2 97
   rat    a =  2 / 1     b =  313 / 464  INT  B: 313 =  313  D: 464 =  2^4 29 A: 928 =  2^5 29
   rat    a =  464 / 313     b =  1 / 2  INT  B: 313 =  313  D: 626 =  2 313 A: 928 =  2^5 29
   rat    a =  117 / 41     b =  5 / 9  INT  B: 205 =  5 41  D: 369 =  3^2 41 A: 1053 =  3^4 13
   rat    a =  9 / 5     b =  41 / 117  INT  B: 205 =  5 41  D: 585 =  3^2 5 13 A: 1053 =  3^4 13
   rat    a =  8 / 1     b =  37 / 146  INT  B: 37 =  37  D: 146 =  2 73 A: 1168 =  2^4 73
   rat    a =  146 / 37     b =  1 / 8  INT  B: 37 =  37  D: 296 =  2^3 37 A: 1168 =  2^4 73
   rat    a =  27 / 26     b =  872 / 975  INT  B: 1744 =  2^4 109  D: 1950 =  2 3 5^2 13 A: 2025 =  3^4 5^2
   rat    a =  110 / 41     b =  13 / 22  INT  B: 533 =  13 41  D: 902 =  2 11 41 A: 2420 =  2^2 5 11^2
   rat    a =  22 / 13     b =  41 / 110  INT  B: 533 =  13 41  D: 1430 =  2 5 11 13 A: 2420 =  2^2 5 11^2
   rat    a =  37 / 12     b =  48 / 73  INT  B: 576 =  2^6 3^2  D: 876 =  2^2 3 73 A: 2701 =  37 73
   rat    a =  73 / 48     b =  12 / 37  INT  B: 576 =  2^6 3^2  D: 1776 =  2^4 3 37 A: 2701 =  37 73
   rat    a =  113 / 102     b =  24 / 25  INT  B: 2448 =  2^4 3^2 17  D: 2550 =  2 3 5^2 17 A: 2825 =  5^2 113
   rat    a =  25 / 24     b =  102 / 113  INT  B: 2448 =  2^4 3^2 17  D: 2712 =  2^3 3 113 A: 2825 =  5^2 113
   rat    a =  111 / 8     b =  10 / 27  INT  B: 80 =  2^4 5  D: 216 =  2^3 3^3 A: 2997 =  3^4 37
   rat    a =  27 / 10     b =  8 / 111  INT  B: 80 =  2^4 5  D: 1110 =  2 3 5 37 A: 2997 =  3^4 37
   rat    a =  1107 / 169     b =  1 / 3  INT  B: 169 =  13^2  D: 507 =  3 13^2 A: 3321 =  3^4 41
   rat    a =  3 / 1     b =  169 / 1107  INT  B: 169 =  13^2  D: 1107 =  3^3 41 A: 3321 =  3^4 41

C++ with GMP

int main()
{
  mpz_class bound = 3370;
  for( mpz_class a = 3; a <= bound; ++a){
  for( mpz_class d = 2; d < a; ++d){
  for(mpz_class b = 1; b < d; ++b){
   mpz_class m1 = d * d * d * d - a * b * b * b ;
     mpz_class m2 =   a * a * a * b - d * d * d * d;
    mpz_class m = m1 * m2;

    if( m > 0 && mp_SquareQ(m)  && mp_three_GCD(a,b,d) == 1 && (!mp_SquareQ(b) || !mp_SquareQ(a)) )
    { 
       mpz_class g = mp_GCD( a,d );
     mpz_class h = mp_GCD( b,d );
       cout << "   rationals    a =  " << a/g << " / " << d / g ;
           cout << "     b =  " << b/h << " / " << d / h ;
       cout << "  integers  B: " << b  << " = " << mp_Factored(b)  << "  D: " << d  << " = " << mp_Factored(d)  << " A: " << a << " = " << mp_Factored(a);

       cout << endl;
    }
  }}}


    return 0 ;
}