Product of primes less than or equal to n

In summary, the conversation discusses a problem where it is necessary to show that the product of primes less than or equal to n is less than or equal to the product of primes between n and 2n. The person asking for help tried to use PNT but was not successful. They also mention having trouble with tex commands and provide a corrected version. However, it is pointed out that the original statement is not always true, with counter-examples given.
  • #1
R.P.F.
211
0

Homework Statement



I'm working on a problem and as long as I can show that
[tex] \prod_{p\leq n}p \leq \prod_{n<p\leq 2n}p [/ tex]
then I'm done. But I'm having trouble with this..Can someone help? :tongue:

Homework Equations


The Attempt at a Solution



I tried to use PNT but could not solve it..
EDIT: and maybe someone could also tell why my tex commands did not work out..?
 
Physics news on Phys.org
  • #2
I can't really tell what you were going for with the tex. can you try to write it out with regular text and maybe then I can straighten out the tex?
 
  • #3
R.P.F. said:

Homework Statement



I'm working on a problem and as long as I can show that
[tex] \prod_{p\leq n}p \leq \prod_{n<p\leq 2n}p [/tex]
then I'm done. But I'm having trouble with this..Can someone help? :tongue:

Homework Equations





The Attempt at a Solution



I tried to use PNT but could not solve it..
EDIT: and maybe someone could also tell why my tex commands did not work out..?

I corrected your tex. There was an extra space between the final "/" and "tex".
 
  • #4
Petek said:
I corrected your tex. There was an extra space between the final "/" and "tex".

And I fail to see the obvious... :( I guess it's bed time. Really cool problem though! (not that I can solve it)
 
  • #5
So, it's not always true. Not true for n less than 8, and not true again for n=18 (510,510 and 392,863). I'm supposing it's not true for other values of n, but I think this is sufficient counter-example to rethink the situation.
 

1. What is the "Product of primes less than or equal to n"?

The "Product of primes less than or equal to n" refers to the multiplication of all prime numbers that are less than or equal to a given number n. For example, if n=10, the product would be 2x3x5x7=210.

2. Why is the "Product of primes less than or equal to n" important?

The "Product of primes less than or equal to n" is important because it helps in determining the prime factorization of a number. This is useful in various mathematical applications, such as finding the greatest common divisor and simplifying fractions.

3. How do you calculate the "Product of primes less than or equal to n"?

To calculate the "Product of primes less than or equal to n", you will need to find all the prime numbers that are less than or equal to n and then multiply them together. This can be done through trial and error or by using a prime factorization calculator.

4. Is there a formula for calculating the "Product of primes less than or equal to n"?

Yes, there is a formula for calculating the "Product of primes less than or equal to n", known as the Primorial function. It is denoted as n# and is defined as the product of all primes less than or equal to n. For example, 6# = 2x3x5 = 30.

5. What are some practical applications of the "Product of primes less than or equal to n"?

The "Product of primes less than or equal to n" has various practical applications in mathematics and computer science. It is used in cryptography, specifically in RSA encryption, and in algorithms for finding the shortest path in graph theory. It is also used in the Chinese remainder theorem and in calculating the number of divisors of a number.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
19
Views
4K
Replies
23
Views
1K
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
16
Views
2K
Back
Top