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Source. P. Erdős, Some unsolved problems, Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221--254, printed pp. 238--239 / PDF pp. 18--19 of the archive's scan. The source was checked visually. The historical statement and the imported formulation are distinguished below.

What the 1961 page says

Following the five-variable indefinite-form discussion, printed p. 239 reads: "It is not known if for every irrational $\alpha$ and $\epsilon>0$ the inequalities

$$\left|(x^2+y^2)\alpha-z^2\right|<\epsilon \qquad\text{and}\qquad \left|x^2+y^2-z^2\alpha\right|<\epsilon$$

are solvable in integers. The case $\alpha=\sqrt2$ is also undecided." The first sentence prints neither $\alpha>0$ nor positivity of every coordinate, and it does not explicitly exclude $(0,0,0)$. Taken literally without a nonzero condition, that triple satisfies both inequalities. The preceding indefinite-form context motivates a nontrivial positive-parameter reading, but it does not insert missing words into the printed sentence.

The imported Problem 496 instead retains only the second inequality, quantifies over all irrational real $\alpha$, and requires all three coordinates to be positive integers. Its exact statement is preserved on the problem page. These are traceable formulations, not a silent replacement of one by the other.

The imported formulation fails at every negative irrational $\alpha$: for positive integers, $|x^2+y^2-\alpha z^2|=x^2+y^2+|\alpha|z^2\ge2+|\alpha|$. Boris Alexeev's Lean development Erdos496.lean proves this negation at $\alpha=-\sqrt2$, $\epsilon=1$.

Positive-parameter formulation

For each irrational $\alpha>0$, the all-positive-coordinate conclusion for the second inequality follows from the distinct published Margulis Banach Center Theorem 1 and the complete coordinate transfer. Its source theorem is used as an explicitly stated external input; its homogeneous-dynamics proof is not reconstructed here.

Bears on. Problem 496.