# Show that n is either a prime or the product of two primes

• Shackleford
In summary: I'm still trying to get used to the pace.In summary, the conversation discusses a mathematical problem which states that for an integer n > 1, if no prime number less than or equal to the cubic root of n divides n, then n is either a prime number or the product of two prime numbers. The solution involves assuming the contrary, that n has three prime factors, and showing that this leads to a contradiction. The hint suggests considering the possibility that one of the prime factors is less than the cubic root of n. The conversation also touches on the topic of multitasking and the importance of getting enough rest.
Shackleford

## Homework Statement

Assume that n > 1 is an integer such that p does not divide n for all primes ≤ n1/3. Show that n is either a prime or the product of two primes. (Hint: assume to the contrary that n contains at least three prime factors. Try to derive a contradiction.)

## Homework Equations

Divisibility, etc.

## The Attempt at a Solution

Assume that there exists three primes such that n = p1p2p3.

I suspect that you need to somehow show that there is a prime less than the cubic root that does divide n, but I'm not sure how to show it.

Shackleford said:

## Homework Statement

Assume that n > 1 is an integer such that p does not divide n for all primes ≤ n1/3. Show that n is either a prime or the product of two primes. (Hint: assume to the contrary that n contains at least three prime factors. Try to derive a contradiction.)

## Homework Equations

Divisibility, etc.

## The Attempt at a Solution

Assume that there exists three primes such that n = p1p2p3.

I suspect that you need to somehow show that there is a prime less than the cubic root that does divide n, but I'm not sure how to show it.

It's difficult to give a hint without giving away the answer! That's a hint!

wabbit
Edit : PeroK is right.

wabbit said:
Well, if so then p1, p2, and p3 each divide n, so what do you conclude ?

Well, I thought that I had to show through some clever means that one of the factors is less than the cubic root.

Come on, think a little bit. Or rather, look - there's no clever trick, it's all here. Go back to the statement of the problem, what does it say ?

wabbit said:
Come on, think a little bit. Or rather, look - there's no clever trick, it's all here. Go back to the statement of the problem, what does it say ?

Okay. I assume that it's three distinct prime factors and not the actual cubic root for some integer, e.g. 53 = 125. In that case, it's obviously contradicted. However, I can see that it implies that at least one of the primes is less than the cubic root. If they were larger, then you wouldn't get n.

Shackleford said:
Okay. I assume that it's three distinct prime factors and not the actual cubic root for some integer, e.g. 53 = 125. In that case, it's obviously contradicted. However, I can see that it implies that at least one of the primes is less than the cubic root. If they were larger, then you wouldn't get n.

##125 = 5^3## is not a contradiction to what's proposed.

PeroK said:
##125 = 5^3## is not a contradiction to what's proposed.

To the condition that p does not divide n for all primes ≤ n1/3? If not, then that's why I assume that it's three distinct primes.

Shackleford said:
To the condition that p does not divide n for all primes ≤ n1/3? If not, then that's why I assume that it's three distinct primes.

For ##n = 125## the primes ##\le n^{1/3}## are ##2, 3## and ##5##. ##125## is divisible by one of these (##p =5##) so does not meet the criteria for ##n##.

It says that the prime numbers that divide n are necessarily above its cubic root.

PeroK said:
For ##n = 125## the primes ##\le n^{1/3}## are ##2, 3## and ##5##. ##125## is divisible by one of these (##p =5##) so does not meet the criteria for ##n##.

Ah, yes. That's right. I was going the wrong way.

geoffrey159 said:
It says that the prime numbers that divide n are necessarily above its cubic root.

So I was correct in post #6?

You must be tired and don't see it. As people said, there is very little room between the question and giving you the answer.

geoffrey159 said:
You must be tired and don't see it. As people said, there is very little room between the question and giving you the answer.

Well, that and I am multitasking at work, so I'll probably see it when I get home.

Shackleford said:
Well, that and I am multitasking at work, so I'll probably see it when I get home.

wabbit said:

And that I should probably stop going to bed after midnight.

geoffrey159
Shackleford said:
And that I should probably stop going to bed after midnight.
Ah yes I should try that too some time

I must've read the problem too quickly. A product of three distinct primes would give you an integer larger than n because the primes that do divide n are larger than the cubic root. I sure hope that's right.

Yes.

wabbit said:
Yes.

Thanks for the help and sorry for the brainfart. This is actually a four-week class, so it's moving right along.

## What does it mean to "show that n is either a prime or the product of two primes"?

This statement means that we need to prove that a given number, n, is either a prime number (can only be divided by 1 and itself) or the product of two prime numbers (two smaller numbers that when multiplied together equal n).

## How do you determine if a number is prime?

A number is prime if it is only divisible by 1 and itself. This means that there are no other smaller numbers that can evenly divide into the given number.

## What is the process for finding the prime factors of a number?

The process for finding the prime factors of a number is known as prime factorization. This involves breaking down a number into its prime factors, which are smaller prime numbers that can be multiplied together to get the original number. For example, the prime factors of 12 are 2 and 3, since 2 x 3 = 6, and 6 x 2 = 12.

## Why is it important to show that a number is either a prime or the product of two primes?

This is important because it helps us understand the fundamental building blocks of numbers. It also allows us to identify patterns and relationships between numbers, and can be used in various mathematical calculations and proofs.

## Can a number be both a prime and the product of two primes?

No, a number cannot be both a prime and the product of two primes. A prime number, by definition, can only be divided by 1 and itself, while the product of two primes requires at least three factors. Therefore, a number cannot satisfy both criteria simultaneously.

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