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Aaronson's notes discuss finding $p$ and $q$ if we know $\phi(N)$ by solving the quadratic equation $X^2-(N-\phi(N)+1)X+N=0$ whose roots are $p$ and $q$. This only works if $N$ is the product of two distinct primes (which is the case in most applications of interest) and if we know $\phi(N)$ exactly.
What doesn't often get mentioned about RSA and ...
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