Why does the series 1/n! diverge in the p-adic metric?

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Discussion Overview

The discussion revolves around the divergence of the series 1/n! in the p-adic metric. Participants explore the implications of p-adic valuation and the behavior of factorials in relation to prime factorization, particularly focusing on how these concepts affect the limit of the series.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions why the limit of 1/n! in the p-adic metric does not equal 0, suggesting it is greater than 1.
  • Another participant proposes finding a lower bound on how many times p divides n! as a potential approach to understanding the divergence.
  • Some participants express uncertainty about how to demonstrate the relationship between the prime factorization of n and the p-adic valuation of n!.
  • A participant suggests considering the subsequence 1/p!, 1/(p^2)!, 1/(p^3)! to analyze the p-adic valuation of these terms.
  • There is a claim that the p-adic valuation of 1/(p^r)! is at least 1, though this is challenged by another participant who suggests that the valuation must be greater than 1.
  • Participants discuss the implications of how many times p divides factorials, with examples provided for specific values like 4! and 8!.

Areas of Agreement / Disagreement

Participants express differing views on the p-adic valuation of factorials and its implications for the series' divergence. There is no consensus on the exact relationship or the correct valuation, indicating ongoing debate.

Contextual Notes

Participants acknowledge the complexity of relating the p-adic valuation to the factorial function and the series in question, with some steps and assumptions remaining unresolved.

Ed Quanta
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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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Seems obvious to me -- can you find a lower bound on how many times p goes into n!?
 
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?
 
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?
 
The only answer to that question that makes sense to me is 1
 
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!)?
 
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!?
 

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