Abstract
Let P +(m) denote the greatest prime factor of the positive integer m. Improving and simplifying work of Dartyge [3] we show that $$|\{n \leq x: P^+(n^2 + 1) < x^{\alpha}\}| \gg x.$$ for $$\alpha > \frac{4}{5}$$ . Here $$\frac{4}{5}$$ improves on the previous exponent $$\frac{149}{179} \approx 0.8324$$ .
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