# Difficult number theory problem proofs

• pat devine
In summary, the professor and TA are offering extra credit for a problem that is not a mathematical proof, but a logical one. There is no upper limit on the density of primes.
pat devine
The following is a repost from 2008 from someone else as there was no solution offered or provided I thought id post one here
Homework Statement neither my professor nor my TA could figure this out. so they are offering fat extra credit for the following problem

Let n be a positive integer greater than 1 and let p1,p2,...,pt be the primes not exceeding n.

show that p1p2...pt<4^n- An mathematically rigourous proof of this would take a long time but I would suggest the following outline which is logically correct I believe:

First we need to consider a lowest bound value of n and for that purpose is can be stated as pt above

so we want to show that p1*p2*p3...*pt < 4^pt

consider the natural log of both sides
log (p1*p2...) = log(4^pt)
consider the left hand side
log(p1*p2*p3...) = log p1+ logp2 + log p3 ...

Now consider the prime number theorem which can be restated such that:
logp1+logp2+logp3...+logpt ~ pt

another way to think about using a well known restatement of the prime number theorem is that the density of primes at any point n (ie the approximate prime gap) is approximately equal to log n.

eg logp1 is ~ the gap between p1 and p2
log22 is ~ the gap between p2 and p3 and so on

therefore is you sum up all the gaps between the first n primes the sum will be (approx.) the value of the nth prime pn.
Therefore the Left hand side = pt (approx within the order +/- 1% for large p)
Now the right hand side log(4^pt) = pt* log 4 = 1.386 * pt
Therefore this proves the original statement since pt < 1.386 * pt

Pat

Last edited by a moderator:
(I removed the boldface and moved the thread to the homework section)

That approach suggests that the inequality is true for large numbers, but it is not a mathematical proof. Note that locally the ratio between the two sides can decrease, e.g. between 101 and 103 (both are prime, and 103 > 16).
You'll need some upper limit on the density of primes, probably together with a manual check of the smaller numbers (this part is unproblematic).

Hello and thanks for that. I know that it's not a a proof but I think it provides a framework to achieve one. In any event it's nothing of any value. I agree with your suggestions however. I guess I was trying to find some proveable number theory problems and found this one. I suspect without basis that many prime number conjectures are true but not proveable. If you have any difficult but proveable suggestions I'd like to hear them.
Many thanks for replying it's my first day here and it's good to get feedback.
P

I'm sure it is possible to prove that inequality.

I would expect the same inequality to hold for 3n.
I guess for every number w>e there are at most a finite set of elements where the product exceeds wn. Which also means there is (a) no product of primes that exceeds en or (b) there is a largest w>e where the inequality is violated somewhere.

## 1. What is a difficult number theory problem proof?

A difficult number theory problem proof is a mathematical argument that shows the validity of a statement or theorem related to number theory. It often involves complex equations, formulas, and logical reasoning.

## 2. How do you approach solving a difficult number theory problem proof?

The first step in solving a difficult number theory problem proof is to carefully read and understand the statement or theorem that needs to be proven. Then, you can use your knowledge of number theory and various mathematical techniques to break down the problem into smaller, more manageable steps.

## 3. What skills are necessary for successfully solving a difficult number theory problem proof?

A strong foundation in number theory, advanced algebra, and logic is essential for solving difficult number theory problem proofs. Additionally, critical thinking, problem-solving, and attention to detail are important skills to have when tackling these types of problems.

## 4. Are there any tips or strategies for approaching difficult number theory problem proofs?

One helpful tip is to start by simplifying the problem and breaking it down into smaller, more manageable parts. It can also be beneficial to work backwards from the desired conclusion to see what steps are needed to get there. Additionally, trying different approaches and techniques can often lead to a breakthrough in solving the proof.

## 5. How important is it to check your work when solving a difficult number theory problem proof?

Checking your work is crucial when solving any difficult problem, including number theory problem proofs. One small mistake can completely change the outcome of the proof, so it is important to double-check all calculations and steps. It can also be helpful to have someone else review your work to catch any errors that you may have missed.

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