I'm having trouble verifying an inequality that is claimed in "PRIMES is in P" (Annals of Mathematics, 160 (2004), 781-793).
I'll state it in such a way that it can be understood without the paper, but for the curious it is in the middle of the proof of Lemma 4.3.
Assume that $n$ is an integer satisfying $n \geq 3$. Then it is claimed that $$n \cdot \prod_{i=1}^{\lfloor \log^2 n \rfloor} (n^i-1) < n^{\log^4 n}.$$ Here the log functions means the base-$2$ logarithm.
It is easy to verify a somewhat weaker inequality. Namely, each term in the product is at most $n^{\log^2 n}-1$, so we have $$n \cdot \prod_{i=1}^{\lfloor \log^2 n \rfloor} (n^i-1) \leq n \cdot (n^{\log^2 n}-1)^{\log^2 n} < n \cdot (n^{\log^2 n})^{\log^2 n} = n^{\log^4 n +1}.$$ However, try as I might I cannot seem to get rid of the $1$ in the exponent.