Math Problem: Find Three Integers

The following math problem can be solved using PyQBPP.

Problem

Find integers $x$, $y$, $z$ that satisfy:

\[\begin{aligned} \frac{1}{x}+\frac{1}{y}+\frac{1}{z} = 1\\ 1 < x < y < z \end{aligned}\]

PyQBPP program

Since PyQBPP can handle polynomial expressions, we first rewrite the constraints. Multiplying both sides of the first constraint by $xyz$ yields:

\[xy+yz+zx - xyz = 0\]

The strict inequalities $x<y<z$ can be encoded as

\[\begin{aligned} 1 &\leq y-x \\ 1 &\leq z-y \end{aligned}\]

For the condition $1<x$, the program simply uses 1 as the lower bound of $x$. When $x=1$, $xy+yz+zx-xyz=y+z>0$, so there is no solution with $x=1$, and the lower bound 1 gives the same result.

The following PyQBPP program makes $x$, $y$, and $z$ native integer variables, declares these expressions as constraints with qbpp.cons(), and solves the problem using the Exhaustive Solver:

import pyqbpp as qbpp

x = qbpp.var("x", integer=(1, 10))
y = qbpp.var("y", integer=(1, 10))
z = qbpp.var("z", integer=(1, 10))

c1 = qbpp.cons(x * y + y * z + z * x - x * y * z == 0)
c2 = qbpp.cons(y - x >= 1)
c3 = qbpp.cons(z - y >= 1)

f = c1 + c2 + c3
f.simplify()
solver = qbpp.ExhaustiveSolver(f)
result = solver.search(best_energy_sols=0)

for sol in result.sols:
    print(f"(x,y,z) = ({sol(x)}, {sol(y)}, {sol(z)})")

The three constraints are declared as c1, c2, and c3, and combined into a single expression f. The energy of f is the sum of the squared violations of the constraints, and it is 0 only when all the constraints are satisfied. The Exhaustive Solver examines all $10^3$ combinations of $x$, $y$, and $z$, and returns all solutions with the minimum energy.

Because x, y, and z are handled as integers and the inequalities introduce no auxiliary variables, the same $(x,y,z)$ is never returned more than once.

This program produces the following output:

(x,y,z) = (2, 3, 6)

This indicates that the problem has exactly one solution in the searched range, namely $(x,y,z)=(2,3,6)$.