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I am looking for an upper bound on the number of integers $n<x$ such that $n$ has a prime factor $p>\log(x)^{(1+\delta)}$ such that $p \equiv a \mod b$. Where $a,b$ are fixed and coprime and $0&...

$w=z=x+ 1 =y−1$ provides $wz−xy=w^2−(w−1)(w+ 1) = 1$. Hence if $x,y$ are odd then $w,z$ are even and all four integers are close. Is there elementary example where only $w$ is even and all four ...

There is a question that has been bothering my mind for quite a while now. I will present it and my current thoughts and progress on it. Let the prime factorization of an integer $n$ be $$n = p_1^{...