Daniel Tubbenhauer on p-adic Arithmetic

This is the best, clearest explanation I have ever seen! See this discussion aboout sqrt 2 being "almost an integer" in 7-adic arithmetic, and Daniel's reply:

Try https://doc.sagemath.org/html/en/reference/padics/sage/rings/padics/tutorial.html that you can run here https://sagecell.sagemath.org/

Here is some basic code that is hopefully self-contained:

R = Zp(7, prec = 10, type = 'fixed-mod', print_mode = 'series')
a = R(10390190017)
b = R(7346972688)
a/b

See also Douglas Hofstadter and Bill Gosper, Bill Gosper's post Fractal curves at rational points & similarly recursively-defined functions and his paper on computing with rational approximations:

There is a conection with diophantine approximation using continued fractions isn't there? I tried to get an idea of what a nilsequence and the conditions for the local-global principle are from Wikipedia but it is incomprehensible to me. I am wondering whether there a series of primes p_i for which the p_i-adic representations of sqrt 2, say, are approximated by a series of p_i-adic integers. Bill Gosper worked out a system of arithmetic on (partial) continued fraction expansions  In the abstract he put "Contrary to everybody, this self contained paper will show that continued fractions are not only perfectly amenable to arithmetic, they are amenable to perfect arithmetic." For over a year I have had it on my TODO list to try define a translation of p-adic arithmetic into continued fraction arithmetic and vice-versa. They are both interesting from the point of view of representing rational arithmetic as a functional isomorphism that can be realised on a finite computing machine.

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