Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence. Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients. If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.
Problem
Source: China Nanchang, Jul 28,2021
Tags: algebra, number theory, China
28.07.2021 10:25
Any answer?
28.07.2021 11:12
interesting
29.07.2021 07:52
Bump this.
29.07.2021 09:19
just lte
29.07.2021 09:58
@above Can you submit your Sol with LTE?
29.07.2021 15:14
Anyone got a solution?
29.07.2021 15:50
Bump this again.
29.07.2021 15:53
Don't freaking bump before 24 hours
30.07.2021 07:30
how to use lte?
03.08.2021 19:54
Bump this.
14.08.2021 20:52
Bumping again.
02.09.2021 04:59
Bump again.
02.09.2021 16:07
Scrutiny wrote: The official solution Someone can translate it into English,pls
04.09.2021 07:00
This result is a special case of Strassmann's theorem on p-adic power series. To prove that $v_n$ is divisible by any large power of $p$, we can express $v_n$ as a polynomial with many integer roots. The polynomial is just $\sum_{i=0}^N \binom{x}{i}p^iu_i$ for large $N$.