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ID: INPUT{id.yaml}

Title: >
  Values of the $p$-adic arithmetic-geometric mean

Definition: >
  The $p$-adic arithmetic-geometric mean $\text{agm}_p(a,b)$ of two
  $p$-adic numbers $a$ and $b$ is defined
  as the simultaneous limit $\lim a_n = \lim g_n$ 
  of the sequences $(a_n)$ and $(g_n)$ given by
  $a_0 = a$, $g_0 = b$, 
  $a_{n+1} = (a_n + g_n)/2$, 
  $g_{n+1} = \sqrt{a_n g_n}$. 
  This table lists values of $\text{agm}_p(a,b)$ 
  for certain pairs of integers $a$ and $b$.
  
Parameters:
  p:
    type: Z
    constraints: >
      prime
  a:
    type: Qp
  b:
    type: Qp
    constraints: > 
      $a/b = 1 \mod p$ (mod $16$ for $p=2$)    
       
Comments:
  comment-breaking-symmetry: >
    Due to the symmetry CITE{formula-symmetric} 
    and trivial cases CITE{formula-diagonal}, 
    we only list values 
    for $a < b$.
  
Formulas: 
  formula-symmetric: >
    $\text{agm}_p(a,b) = \text{agm}_p(b,a)$
  formula-diagonal: >
    $\text{agm}_p(a,a) = a$

Programs:

References:

Links:
  MathWorld:
    title: "MathWorld: Arithmetic-Geometric Mean"
    url: https://mathworld.wolfram.com/Arithmetic-GeometricMean.html
  Wiki:
    title: "Wikipedia: Arithmetic-geometric mean"
    url: https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean

Similar tables:
  
Keywords:
  
Tags:
- p-adic
- number theory
- elliptic curves

Data properties:
  type: R
  complete: no

Display properties:
  number-header: $\text{agm}_p(a,b)$

Numbers: INPUT{numbers.yaml}

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Values of the $p$-adic arithmetic-geometric mean

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Numbers

$p$

$a$

$b$

$\text{agm}_p(a,b)$

INPUT{numbers.yaml} (not shown in preview)

Definition

The $p$-adic arithmetic-geometric mean $\text{agm}_p(a,b)$ of two $p$-adic numbers $a$ and $b$ is defined as the simultaneous limit $\lim a_n = \lim g_n$ of the sequences $(a_n)$ and $(g_n)$ given by $a_0 = a$, $g_0 = b$, $a_{n+1} = (a_n + g_n)/2$, $g_{n+1} = \sqrt{a_n g_n}$. This table lists values of $\text{agm}_p(a,b)$ for certain pairs of integers $a$ and $b$.

Parameters

$p$

— integer (prime
)

$a$

— p-adic number

$b$

— p-adic number ($a/b = 1 \mod p$ (mod $16$ for $p=2$)

)

Formulas

(1)

$\text{agm}_p(a,b) = \text{agm}_p(b,a)$

(2)

$\text{agm}_p(a,a) = a$

Comments

(3)

Due to the symmetry (1) and trivial cases (2), we only list values for $a < b$.

Links

[1]

MathWorld: Arithmetic-Geometric Mean

[2]

Wikipedia: Arithmetic-geometric mean

Data properties

Entries are of type: real number

Table is complete: no