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Consider the expression $$P(x):=\text{lcm}_{\text{$p$ prime $\leq x$}} (p^2-1).$$ As the prime factors of $P(x)$ are $<x$, for every prime $q<x$ its highest power in $P(x)$ must also divide some $p^2-1$ and is therefore $\leq p^2-1 \leq x^2-1$. Thus we have an obvious upper bound of the form $P(x)<(x^2-1)^{\pi(x)}$, that is $\log P(x)=O(x)$.

Can one do better than this? What if instead of $p^2-1$ we had $p+1$ or $p-1$?

[I am thinking of substantial improvements, such as $O(x\log \log x /\log x)$ or $O(x/\log x)$. The above argument with more care leads to the upper bound $\prod_{p \leq x} (p^2-1)$, but its log is still an $O(x)$.]

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  • $\begingroup$ I forgot to add that with some powerful tools such as Elliott-Halberstam one can do better than this, but I need unconditional answers. $\endgroup$
    – user68136
    Aug 2, 2014 at 10:42
  • $\begingroup$ LCM of $p-1$ is tabulated at oeis.org/A058254 (but with no links or information about bounds). Similarly, LCM for $p+1$ at oeis.org/A085272 $\endgroup$ Aug 3, 2014 at 0:18

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You are probably aware of the following, but here are some heuristics that could be useful to someone. I am not entirely sure how much of it can be made formal.

If $q$ is a small prime, then $P(n)$ will be divisible by $q$ for large $n$, because for large $n$ you will always be able to find a prime $p \equiv \pm 1 \pmod{q}$, by Dirichlet's theorem. In other words, $P(n)$ must be divisible by lots of small primes. However, I claim that $P(n)$ will not be divisible by very many large primes.

More precisely, let $q$ be a prime number larger than $M := \frac{n}{\sqrt{\log(n)}}$. Then the probability that either $2q-1$ or $2q+1$ is prime is roughly $\frac{c}{\log(n)}$ for some small computable constant $c$. And the same holds true for $aq\pm1$ for any $a$ with $2 \le a \le \frac{n}{q} < \sqrt{\log(n)}$. So the probability that $q$ divides $P(n)$ is upper bounded by

$$1 - \left(1 - \frac{c}{\log(n)}\right)^{\sqrt{\log(n)}} \approx 1 - e^{\frac{-c}{\sqrt{\log(n)}}} \approx \frac{c}{\sqrt{\log(n)}}$$

Therefore, ignoring factors of $o(1)$ for convenience,

$$P(n) \le n^{\pi(M)} n^{\frac{c(\pi(n) - \pi(M))}{\sqrt{\log(n)}}} = e^{\frac{n}{\sqrt{\log(n)}}} e^{\frac{cn}{\sqrt{\log(n)}}} = e^{\frac{(c+1)n}{\sqrt{\log(n)}}} $$

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