Bonse's Inequality: Estimating Lower Bound on Prime Powers

[a1call]

Hi all,
https://en.m.wikipedia.org/wiki/Bonse's_inequality

It seems to me that the inequality can be true for higher powers (if not any given higher power), for an appropriately higher (lower) bound for "n".

Any thoughts, proofs, counter proofs your insights are appreciated.

In particular, I am interested in estimates (or preferably lower bound) for m for:

$(p-1)# < p^m$
where p is any prime number.

Thank you in advance.

 

[mfb]

There is always a prime between n and 2n (https://en.wikipedia.org/wiki/Bertrand's_postulate]Bertrand's[/PLAIN] postulate, proved 1852), therefore $p_n p_{n-1} p_{n-2} \dots > \frac{1}{2} p_{n+1} \frac{1}{2^2} p_{n+1} \frac{1}{2^3} p_{n+1} \dots$. As long as the product of the remaining primes is larger than 64, the product is larger than $p_{n+1}^3$. That happens for 2*3*5*7, so 2*3*5*7*11*13*17 = 510510 > 6859 = 19³ is the first number where the general proof works, but 2*3*5*7*11 > 13^3 is where the inequality starts being valid.

It should be obvious how to extend that to larger powers.

I'm surprised Bonse's inequality got a special name.

 

[a1call]

Thank you very much mfb.
Typical exemplary reply.