Solve a system of equations mixing real and integer variables

Question

I'm trying to solve a system of equations and inequalities with 5 variables but one of them is real, and the other 4 are integer. How could I tell this to Mathematica? I tried using FindInstance[eq && eq && ineq && ineq && eq, {a, b, c, d, e}, Integers] but I don't know how to specify, let's say, e is a real variable.

My particular case is this set of equations:

$\frac{a(a+1) + b^2(c-1)}{d(d+1) + e^2(c-1)} = 1$, $|b|\leq a$, $|e| \leq d$, $a>0$, $d>0$, $c > 0$,

where all variables are integers except from $c$ which is real.

Could you give me some hints?

Thak you!

Answer 1

Hints

Solve[a (a + 1) + b^2 (c - 1) == d (d + 1) + e^2 (c - 1), c]

Out[1]= {{c -> (-a - a^2 + b^2 + d + d^2 - e^2)/(b^2 - e^2)}}

 FindInstance[
 Abs[b] <= a && Abs[e] <= d && d > 0 && 
  a > 0 && (-a - a^2 + b^2 + d + d^2 - e^2)/(b^2 - e^2) > 0, {a, b, d,
   e}, Integers]

Out[4]= {{a -> 8, b -> -2, d -> 8, e -> -4}}

Calculate c

 (-a - a^2 + b^2 + d + d^2 - e^2)/(b^2 - e^2) /. First[%]

Out[5]= 1