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

 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+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 $b$.+  certain pairs of integers $a$ and $b$. Over $\mathbb{Q}_p$ the equation $g_{n+1}^2+  = a_n g_n$ has two solutions and they lead to different limits, so the iteration+  alone does not determine the value: here $g_{n+1}$ is the square root nearer to+  $a_{n+1}$, that is the one for which $|g_{n+1} - a_{n+1}|_p$ is smaller. This is+  the convention used by PARI, whose $\texttt{agm}$ produced the values below.' Parameters:   p:

   formula-symmetric: $\text{agm}_p(a,b) = \text{agm}_p(b,a)$   formula-diagonal: $\text{agm}_p(a,a) = a$+  formula-step-invariance: '$\text{agm}_p(a,b) = \text{agm}_p\left(\frac{a+b}{2},+    \sqrt{ab}\right)$ for the same choice of root, which is what makes the choice+    checkable: only one of the two satisfies it.' Programs: {} References: {}