Upper bound for least common multiple of the first n positive integers

Question

Goal: Show that $\textit{lcm}(1,...,n)\leq 3^n$.

I have shown that for every integer $n>0$ one has that the $\text{lcm}(1,...,n)=\prod_{p\leq{n}} p^{\lfloor{\log_p{n}}\rfloor}$ where $p$ are primes. Then we get that $\text{lcm}(1,...,n)\leq n^{\pi(n)}$ whereby $\pi(n)$ is prime counting function.

Can someone help me proving this bound? I thought of trying to use the prime number theorem but I seem to miss some logarithmic identities or another inequality to get the result above.

Does this answer your question? Inequality regarding LCM of $1, 2, \ldots, n$. There are also other closely related questions here, e.g., Question regarding $C(n)$ and $B(n)$ in Hanson's proof that $\prod\limits_{p^a \le n} p < 3^n$ and Still unclear why Hanson's proof that $\text{lcm}(1,…,n)<3^n$ is not sufficient to resolve Legendre's Conjecture.. — John Omielan, Sep 01 '21 at 08:49
FYI, a somewhat related question is Reason for LCM of all numbers from 1 .. n equals roughly $e^n$. — John Omielan, Sep 01 '21 at 08:56
Thank you for the quick reply! In these questions Hansons proof of the upper bound is used (it doesn't rely on the prime number theorem) and it sais in the first question you linked, that using the prime number theorem the inequality is "obvious" to show, but I don't get how. Could you help me out? @JohnOmielan — alexalexalex, Sep 01 '21 at 12:17
@alexalexalex To me it is not "obvious" , but the asymptotic expression is surely the key to show this inequality. — Peter, Sep 01 '21 at 17:15
@alexalexalex You're welcome for my fast reply. Note I agree with Peter's comment above since I also don't consider it to be "obvious". Nonetheless, the answer to the question I linked to in my FYI comment above helps to shows one way the Prime Number Theorem can be used, basically as Peter described in his comment. — John Omielan, Sep 01 '21 at 19:09

Upper bound for least common multiple of the first n positive integers

0 Answers0