Largest prime factor
- Tolaso J Kos
- Administrator
- Posts: 867
- Joined: Sat Nov 07, 2015 6:12 pm
- Location: Larisa
- Contact:
Largest prime factor
Find the largest prime factor of: $$\frac{1}{\sum \limits_{n=1}^\infty \frac{2012}{n(n+1)(n+2)(n+3)\dots(n+2012)}}$$
Imagination is much more important than knowledge.
-
- Former Team Member
- Posts: 77
- Joined: Mon Nov 09, 2015 11:52 am
- Location: Limassol/Pyla Cyprus
- Contact:
Re: Largest prime factor
Note that \[ \frac{2012}{k(k+1)\cdots (k+2012)} = \frac{1}{k(k+1)\cdots(k+2011)} - \frac{1}{(k+1)(k+2)\cdots (k+2012)}.\] So the sum \[ \sum_{n=1}^{\infty} \frac{2012}{n(n+1)\cdots (n+2012)}\] is telescopic and equals \(1/(2012)!\). In particular the expression of the question is equal to \(2012!\) and so it is indeed an integer. (The question wouldn't make sense otherwise!) Its largest prime factor is the largest prime in the set \(\{1,2,\ldots,2012\}\). It can be checked that \(2011\) is prime. and so it is the largest prime factor of the expression.
- Tolaso J Kos
- Administrator
- Posts: 867
- Joined: Sat Nov 07, 2015 6:12 pm
- Location: Larisa
- Contact:
Re: Largest prime factor
A relative topic can be found in the link: Series.
The topic is about the evaluation of the series, which indeed telescopes. But I have another way in mind too, with the use of Beta & Gamma functions. Demetres thank you for the solution. It comes from Putnam competition and I think it was quite recent.
The topic is about the evaluation of the series, which indeed telescopes. But I have another way in mind too, with the use of Beta & Gamma functions. Demetres thank you for the solution. It comes from Putnam competition and I think it was quite recent.
Imagination is much more important than knowledge.
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 18 guests