P-Adic Metric: Exploring Divergence in the Series of 1/n!

In summary, the conversation discusses the divergence of the series of 1/n! in the p-adic metric and how the p-adic valuation of n! relates to the number of times p goes into n!. The participants consider the subsequence of 1/(p^r)! and try to find the minimum number of times p must divide it, with the answer being at least 1. They also discuss finding multiples of 2 that divide various factorials.
  • #1
Ed Quanta
297
0
Why does the series of 1/n! diverge in the p-adic metric?In other words, how do I show that the lim of 1/n! (in the p-adic metric) does not equal 0 because it is >1
 
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  • #2
Seems obvious to me -- can you find a lower bound on how many times p goes into n!?
 
  • #3
No, but I am not sure how to show this. I know that every n has a prime factorization, and that as n increases, p will go into n more and more times. But what does this mean?
 
  • #4
COnsider the subsequence 1/p!, 1/(p^2)!, 1/(p^3)! What is the p-adic valuation of 1/(p^r)! at least as great as?
 
  • #5
The only answer to that question that makes sense to me is 1
 
  • #6
No, but I am not sure how to show this. I know that every n has a prime factorization, and that as n increases, p will go into n more and more times. But what does this mean?

So, how does the p-adic valuation of n! relate to the number of times p goes into n!?

How does that relate to the p-adic valuation of 1/(n!)?
 
  • #7
Ed Quanta said:
The only answer to that question that makes sense to me is 1


Eh? (p^r)! how many times at least must p divide this? You can do better than 1, surely? find a multiple of 2 dividing 4! such as 4, one for 8! such as 8, what about 16!?
 

Related to P-Adic Metric: Exploring Divergence in the Series of 1/n!

1. What is divergence in p-adic metric?

Divergence in p-adic metric refers to the behavior of a sequence in the p-adic metric space. A sequence is said to diverge if it does not have a limit in the p-adic metric space.

2. How is divergence in p-adic metric different from divergence in other metric spaces?

The concept of divergence in p-adic metric is unique because the p-adic metric space has different properties compared to other metric spaces. In p-adic metric, the distance between two numbers is determined by the highest power of p that divides their difference. This leads to different behavior of sequences and their convergence or divergence.

3. What are the conditions for a sequence to diverge in p-adic metric?

A sequence in p-adic metric diverges if its terms become arbitrarily large in the p-adic metric space. This can happen if the sequence has a term with a high power of p, which dominates the other terms and makes the sequence diverge.

4. Can a sequence diverge in p-adic metric but converge in other metric spaces?

Yes, a sequence can diverge in p-adic metric but converge in other metric spaces. This is because the behavior of sequences in p-adic metric is based on the properties of p-adic numbers, which are different from real or complex numbers in other metric spaces.

5. What are the applications of studying divergence in p-adic metric?

Studying divergence in p-adic metric has applications in number theory, algebraic geometry, and cryptography. It also has connections to other areas of mathematics, such as fractal geometry and dynamical systems.

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