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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Large prime factors of n²+1

Iwaniec proved (and many people extended) that the number of $n \le x$ for which $n^2+1=P_2$ (product of at most two primes) is $\gg x/\log x$. I am wondering what is known/can be proved for the count …
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