Hyperbolic Diophantine Equations: Application of Euclidean Algorithm?

Question

I'm trying to determine whether or not I can find the integer solutions to $(x+a)$$(x+b)$ $=$ $x(x-1)$ + $x(a-b)$ (with a known $x$ value you choose, i.e. $707$). Plugging in my example value on Wolfram Alpha for $x$ reveals the form of a hyperbola, but can I use http://mathworld.wolfram.com/EuclideanAlgorithm.html to help me generalize solutions for hyperbolic equations of this form?

Presumably plugging in the function into Wolfram Alpha and choosing my integer $x$ value for every solution is not the only way to do this?

Answer 1

$$(x+a)(x+b)=x(x-1) + x(a-b),$$ $$(x+a)(x+b)-x(x-1) + x(a-b)= 0,$$ cancel stuff and $$ (2b+1)x + ab = 0. $$

Introduce a variable $$ c = a + 2x, $$ so that $$ a = c - 2x. $$ Then $ (2b+1)x + ab = 0 $ becomes $$ x + bc = 0, $$ $$ bc = -x. $$ Therefore, find all divisors $d$ of $-x,$ both positive and negative, so that $$ d \in \{ -|x|, \ldots, -1,1 \ldots, |x| \}. $$ For each such $d,$ let $$ c = d, \; \; \; b = -x / d, $$ with $$ a = d - 2x, \; \; b = -x / d. $$

Finding all positive divisors of $|x|$ is a matter of completely factoring $|x|;$ for example, your $707 = 7 \cdot 101.$ The positive divisors are $1,7,101,707,$ and all divisors are $$d \in \{-707, -101,-7,-1,1,7,101,707 \}.$$ For each such $d$, let $a = d - 1414$ and $b = -707/d.$

set<mpz_class> mp_Divisors( mpz_class  i)
{
  set<mpz_class> sofar, more;
    set<mpz_class>::iterator iter;
  sofar.insert(1);
  more.clear();
  string fac;
  fac = " = ";
  mpz_class p = 2;
   mpz_class  temp = i;
  if (temp < 0 )
  {
    temp *= -1;
    fac += " -1 * ";
  }

  if ( 1 == temp) fac += " 1 ";
  if ( temp > 1)
  {
    int primefac = 0;
    while( temp > 1 && p * p <= temp)  // WWWWWWWWWWWWWWWWWWW
    {
      if (temp % p == 0)
      {
        ++primefac;
        if (primefac > 1) fac += " ";
      //   fac += stringify( p) ;
        temp /= p;
        int exponent = 1;
          mpz_class power = p;
          for(iter = sofar.begin() ;  iter != sofar.end(); ++iter)
          {
             more.insert( power * *iter );
          }
        while (temp % p == 0)
        {
          temp /= p;
          ++exponent;
          power *= p;
           for(iter = sofar.begin() ;  iter != sofar.end(); ++iter)
          {
             more.insert( power * *iter );
          }
        } // while p is fac
        if ( exponent > 1)
        {
          fac += "^" ;
          fac += stringify( exponent) ;
        }
         for(iter = more.begin() ;  iter != more.end(); ++iter)
          {
             sofar.insert(  *iter );
          }
          more.clear();
      }  // if p is factor
      ++p;
    } // while p
    if (temp > 1 && primefac >= 1) fac += " ";
  //  if (temp > 1 ) fac += stringify( temp.get_ui())  ;
         if (temp > 1 ) fac +=  temp.get_str()  ;
    if ( temp > 1) {
        for(iter = sofar.begin() ;  iter != sofar.end(); ++iter)
          {
             more.insert( temp * *iter );
          }
        for(iter = more.begin() ;  iter != more.end(); ++iter)
          {
             sofar.insert(  *iter );
          }
          more.clear();
    }
  } // temp > 1
  return sofar;
} // mp_Divisors