Tunnell's theorem expresses the congruent number property of a squarefree $n$ through the numbers of integer solutions of $2x^2 + y^2 + 32z^2 = n$, $2x^2 + y^2 + 8z^2 = n$ ($n$ odd) and $4x^2 + y^2 + 32z^2 = m$, $4x^2 + y^2 + 8z^2 = m$ ($n = 2m$). In all four the coefficients of $x^2$ and $z^2$ are even and positive.

This concept defines the solution set of $a x^2 + y^2 + c z^2 = n$ as the finite set of integer triples with all coordinates in $[-n, n]$ (for $a, c > 0$ and $n \ge 0$ this contains every solution, since each square is at most $n$), and states that for even $a$ and $c$ and odd $n$ its cardinality is even. The reason is the fixed-point-free involution $(x, y, z) \mapsto (x, -y, z)$: a fixed point would have $y = 0$, hence $n = a x^2 + c z^2$ even.

The positivity hypotheses on $a$ and $c$ are not needed for the parity argument; they are what makes the bounded set equal to the full solution set, so that the statement is about Tunnell's counts and not about a truncation of them.