Bonse's Inequality: Estimating Lower Bound on Prime Powers

• A
• a1call
In summary, the conversation discussed Bonse's inequality and its validity for higher powers with an appropriately higher or lower bound for "n". The conversation also touched on estimates and lower bounds for m in the case of ##(p-1)# < p^m## where p is any prime number. One participant mentioned that there is always a prime between n and 2n, as proven by Bertrand's postulate in 1852, and provided a general proof for Bonse's inequality in the case of 2*3*5*7*11*13*17 = 510510 > 6859 = 193. The conversation ended with a participant expressing surprise at the special name given to Bonse's inequality.
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.

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.

Last edited by a moderator:
Thank you very much mfb.

1. What is Bonse's Inequality?

Bonse's inequality is a mathematical concept that allows us to estimate the lower bound on prime powers. It is named after the mathematician Carl Ludwig Bonse, who first proposed it in 1909.

2. How does Bonse's Inequality work?

Bonse's inequality is based on the fact that every composite number can be expressed as the product of prime numbers. By using this fact, we can determine the lower bound on prime powers by looking at the smallest prime factor of a composite number.

3. Why is Bonse's Inequality important?

Bonse's inequality is important because it provides a quick and easy way to estimate the lower bound on prime powers. This information is useful in many areas of mathematics, including number theory and cryptography.

4. What are some applications of Bonse's Inequality?

One application of Bonse's inequality is in the field of cryptography, where it can be used to determine the security of certain encryption algorithms. It is also used in number theory to study the distribution of prime numbers.

5. Are there any limitations to Bonse's Inequality?

Yes, there are limitations to Bonse's inequality. It is only an estimation and does not give an exact value for the lower bound on prime powers. It also assumes that the smallest prime factor of a composite number is a prime number itself, which is not always the case.

Replies
9
Views
1K
Replies
13
Views
2K
Replies
4
Views
3K
Replies
1
Views
793
Replies
3
Views
4K
Replies
4
Views
1K
Replies
8
Views
5K
Replies
5
Views
3K
Replies
1
Views
978
Replies
33
Views
5K