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October 5th, 2017, 03:50 PM   #1
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Prove or disprove

Let n=p*q, where p and q are primes.

Prove (or disprove) that:

n! is divisible by (p!)^q or by (q!)^p
Thanks from zylo

Last edited by skipjack; October 6th, 2017 at 05:18 AM.
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October 6th, 2017, 03:56 AM   #2
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This one is the most interesting.
Everybody knows Wilson's theorem.

I'm going to give 2 examples.

Example 1:

We want to check whether n=13 is prime or not.
We compute n-1=13-1=12
12 is even hence 12=2*6

We compute (2!)^6 mod 13
We compute (6!)^2 mod 13

What will happen if 13 is prime?

Example 2:

We want to check whether n=15 is prime or not.
We compute n-1=15-1=14
14 is even hence 14=2*7

We compute (2!)^7 mod 15
We compute (7!)^2 mod 15

What will happen if 15 is not prime?

Those 2 examples shows that there is room for a new theorem for primality.

Instead of computing (n-1)! mod n we break (n-1) is 2 and we compute separately
the 2 parts as described above.

If you could write in Latex and rewrite what I wrote, then will come the time to prove it.

Last edited by skipjack; October 6th, 2017 at 12:25 PM.
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October 6th, 2017, 04:11 AM   #3
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Try this by factorizing n-1 (other than 2*stuff) in 2 factors and see what will happen.
Try it for the first 100 natural numbers (from 1 to 100).
That is only the first step.
There are more to come.

I will come back tomorrow because I have something to do today.

Good work!
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October 6th, 2017, 11:10 AM   #4
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Are you having a good time having a discussion with yourself?

Did you really tell yourself "good work!" ?

Maybe a bit of professional help is called for?
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October 9th, 2017, 05:58 AM   #5
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Too hard maybe.
172 views! and no serious answer yet.
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October 10th, 2017, 04:02 AM   #6
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Quote:
Originally Posted by romsek View Post
Are you having a good time having a discussion with yourself?

Did you really tell yourself "good work!" ?

Maybe a bit of professional help is called for?
No one here is professional.
I said it a long time ago.
What you have here in this forum is that some people claim wrongly that they are professional because they "master" technical tricks. That's it.
You do not need high IQ just a memorizing such techniques.

I know personally many dumb guys who became maths teachers.

Last edited by skipjack; October 10th, 2017 at 04:27 AM.
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October 10th, 2017, 04:04 AM   #7
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Now!
Adios
The forum is dying because of dumb guys like Romstek, Varchie, Joppy and so on.
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October 10th, 2017, 04:39 AM   #8
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Quote:
Originally Posted by mobel View Post
Now!
Adios
The forum is dying because of dumb guys like Romstek, Varchie, Joppy and so on.
You pose problems you already know the answer to, then when nobody answers your posts (because, you know, you already have the answer...) you call us dumb and quit for a while, only to return weeks or months later and do the same thing all over again.

This forum isn't dying, it's just not what you want it to be... I can't speak for everyone, but generally speaking, I've found that most people here are not interested in stroking people's egos and are much more interested in helping people out with real study problems that they can't solve. In my particular case, I also have less time to dedicate to learning new mathematics than I used to and I often don't know the answers to the problems you're posing. I'm much more interested in helping people with simple physics, numerical techniques or elementary mathematics problems and I dip in where I can to help out.

If you post something and nobody responds, maybe you should ask yourself why no one is responding rather than just moaning about the forum.
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October 10th, 2017, 10:26 AM   #9
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(3$\displaystyle \cdot$2)!=6$\displaystyle \cdot$5$\displaystyle \cdot$4$\displaystyle \cdot$3$\displaystyle \cdot$2$\displaystyle \cdot$1 = (6!-3!)(3!) which is divisible by (3!)$\displaystyle ^{2}$.

The generalization is obvious.

Thanks mobel for very interesting thread.
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October 10th, 2017, 01:56 PM   #10
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Quote:
Originally Posted by zylo View Post
(3$\displaystyle \cdot$2)!=6$\displaystyle \cdot$5$\displaystyle \cdot$4$\displaystyle \cdot$3$\displaystyle \cdot$2$\displaystyle \cdot$1 = (6!-3!)(3!)
This is not true: $(3 \cdot 2)! = 720$ while $(6! - 3!)(3!) = 4284$.
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