Problems for v

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xytunghoanh

25 posts

xytunghoanh

#1 h Apr 5, 2025, 1:28 PM

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Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>

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whwlqkd

97 posts

whwlqkd

#2 h Apr 5, 2025, 2:03 PM

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What about this one?
Find all positive integer pair $(k,n)$ that satisfies $k!=(2^n-1)(2^n-2)(2^n-4)(2^n-8)…(2^n-2^{n-1})$

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Sadigly

147 posts

Sadigly

#3 h Apr 5, 2025, 2:50 PM

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whwlqkd wrote:

What about this one?
Find all positive integer pair $(k,n)$ that satisfies $k!=(2^n-1)(2^n-2)(2^n-4)(2^n-8)…(2^n-2^{n-1})$

My man, he is junior,and asks for easy problems,why IMO question?

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whwlqkd

97 posts

whwlqkd

#4 h Apr 5, 2025, 2:55 PM

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What is the aops scale of easy question?
(6=IMO P1, 7.5=P2, 4=Junior Balkan P1, 5=Junior Balkan P2)

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mathnerd_101

1491 posts

mathnerd_101

#5 h Apr 5, 2025, 2:56 PM

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I'm pretty new to $v_p(n)$ stuff, so here's a bold shot. Someone tell me where I went wrong.

Bogus Solution

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AshAuktober

993 posts

AshAuktober

#6 h Apr 5, 2025, 2:58 PM

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Well the first equation you used is incorrect. Since $\nu_p$ extracts an exponent, you can expect it to behave somewhat like a logarithm.
In particular, $\nu_p(mn)=\nu_p(m)+\nu_p(n).$

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Maximilian113

549 posts

Maximilian113

#7 h Apr 5, 2025, 2:59 PM

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yea the IMO is made easier by knowing you have to use v_p, but i guess its still not the easiest

Try:
https://artofproblemsolving.com/community/c6h3041815
https://artofproblemsolving.com/community/c6h2317634s1_ncg_returns
https://artofproblemsolving.com/community/c6h220915p1225024

Suppose that $m$ and $k$ are non-negative inegers and $p=2^{2^m}+1$ is a prime number. Prove that $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1 \pmod {p^{k+1}}.$

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mathnerd_101

1491 posts

mathnerd_101

#8 h Apr 5, 2025, 2:59 PM

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Ohhh... oops! Looks like Stack Exchange is wrong... lol. I'll think about it some more, then.

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AshAuktober

993 posts

AshAuktober

#9 h Apr 5, 2025, 3:01 PM • 2 Y

Y by scannose, whwlqkd

Here's an easy problem you can try.
Let $n$ be a positive integer. Evaluate $\nu_3(2^{3^n}+1).$

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whwlqkd

97 posts

whwlqkd

#10 h Apr 5, 2025, 3:11 PM

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lol…it is so easy
Click to reveal hidden text

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mathnerd_101

1491 posts

mathnerd_101

#11 h Apr 5, 2025, 3:13 PM • 1 Y

Y by whwlqkd

Solution

Edit: Thanks so much. This was actually a really nice introduction to p-adic for me.

Also @above, please note there's people not as smart as you. What may seem easy to you may not be as easy to others. Please do be considerate and kind. Thanks C:

This post has been edited 2 times. Last edited by mathnerd_101, Apr 5, 2025, 3:14 PM

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AshAuktober

993 posts

AshAuktober

#12 h Apr 5, 2025, 3:30 PM

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mathnerd_101 wrote:

Solution

Edit: Thanks so much. This was actually a really nice introduction to p-adic for me.

Also @above, please note there's people not as smart as you. What may seem easy to you may not be as easy to others. Please do be considerate and kind. Thanks C:

Hmm try this without LTE.

Also to be tried:
If $a,b,c, \frac ab +\frac bc+\frac ca \in \mathbb{N}$ , prove that $abc$ is a perfect cube.

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Sadigly

147 posts

Sadigly

#13 h Apr 5, 2025, 3:41 PM

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Prove that for all $n>1$ , the expression $\mathbb{H}_n=\sum_{i=1}^{n}\frac{1}{i}$ is not a integer

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xytunghoanh

25 posts

xytunghoanh

#14 h Apr 5, 2025, 3:41 PM

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Oh, I'm new to $v_p(n)$ . I need some articles, problems to practice and learn.

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mathnerd_101

1491 posts

mathnerd_101

#15 h Apr 5, 2025, 3:48 PM

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@above I used this one here. It had helpful practice problems which I enjoyed.

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AshAuktober

993 posts

AshAuktober

#16 h Apr 5, 2025, 3:59 PM

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Hmm, the one you sent doesn't have a proof, does it? https://cjquines.com/files/liftingtheexponent.pdf might be better.
(Note that proving LTE is a good exercise so try it solo.)

This post has been edited 1 time. Last edited by AshAuktober, Apr 5, 2025, 4:00 PM

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