P-adic valuation expression for a given natural number

In summary, the author derives a formula for ##v_p(n)## which is based on the sum of the digits in the base-p expansion of n.
  • #1
DaTario
1,039
35
TL;DR Summary
Hi All, is there a closed expression that yields the p-adic valuation of a natural number n ?
Hi All,

If p is a prime, the p-adic valuation, ## v_p(n) ##, of a positive natural number is defined as the highest power of p that divides n. For instance, ##v_3(45) = 2##.
My question is: Is there a mathematical expression for ##v_p(n)## ?
 
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  • #2
doubtful - primes are too scattered around
 
  • #3
Thank you, mathman. Could you please elaborate a bit more on this comment?
 
  • #4
I have just found this paper:
https://arxiv.org/ftp/arxiv/papers/1907/1907.11902.pdf
where the author derives a formula for ##v_p(n)##. The approach goes like this:

##v_p (n) = v_p (n!) – v_p ((n –1)!)##
##= (n – s_p (n))/(p –1) – (n –1– s_p (n–1))/(p –1) ##
##= (1 – s_p (n) + s_p (n–1))/(p –1)##
##= (1 – Δ s_p (n–1))/(p –1). ##

where ##s_p (n)## denotes the sum of the digits in the base-p expansion of n.

Example: ##v_3(36) = 2 ##
##36 = 1 . 9 + 1 . 27 \Rightarrow s_3(36) = 1 + 1 = 2##
##35 = 2 . 1 + 2 . 3 + 1 . 27 \Rightarrow s_3(35) = 2 + 2 + 1 = 5##
Thus,
##v_3(36) = \frac{1 - \Delta s_3(36) + \Delta s_3(35)}{3-1} = \frac{1 - 2 + 5}{2} = \frac{4}{2} = 2 ##.

Thak you all.
 
Last edited:
  • #5
Editing the last post: In the last equation there is no ##\Delta## anymore.
 
Last edited:
  • #6
is there some formula for the sum of the digits in the base p expansion? this is something that seems to require an iterative application of the euclidean algorithm.
 
  • #7
Yes, mathwonk, it seems to require such algorithmic approach. Do you think using instead functions as Floor[ ], Ceiling[ ] and FractionalPart[ ], to implement ##v_p(n)##, will demand the same algorithmic structure?
 
  • #8
I found a simple and alternative formula for ##v_p(n)##, given by:
$$
v_p(n) = \lfloor \log_p (n) \rfloor - \sum_{j=1}^{\lfloor \log_p (n) \rfloor} \left \lceil \left \{ \frac{n}{p^j}\right \} \right\rceil,
$$

But it does not seem to be of any practical use.
 
  • #9
DaTario said:
Thank you, mathman. Could you please elaborate a bit more on this comment?
No deep study. Just a gut feeling, since prime factoring seems to be "random" as a function of n.
 
  • #10
Thank you, mathman.
 

1. What is a P-adic valuation expression?

A P-adic valuation expression is a mathematical expression that represents the highest power of a prime number, P, that divides a given natural number. It is commonly used in number theory to analyze the properties of prime numbers and their relationships with other numbers.

2. How is a P-adic valuation expression calculated?

The P-adic valuation expression for a natural number, n, is calculated by finding the largest integer, k, such that P^k is a factor of n. This can be done by dividing n by P repeatedly until the remainder is no longer divisible by P. The value of k is then the P-adic valuation of n.

3. What is the significance of P-adic valuation in number theory?

P-adic valuation is important in number theory because it helps us understand the properties of prime numbers and their relationships with other numbers. It can also be used to prove certain theorems and solve mathematical problems related to prime numbers.

4. Can P-adic valuation be applied to numbers other than primes?

Yes, P-adic valuation can be applied to any natural number. However, it is most commonly used for prime numbers because they have unique properties that make them useful in number theory.

5. How does P-adic valuation relate to other mathematical concepts?

P-adic valuation is closely related to concepts such as prime factorization, greatest common divisor, and modular arithmetic. It is also used in the study of algebraic number theory and has applications in cryptography and coding theory.

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