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Does anyone know a website where I can enter a prime base and a rational and then get the $ p $-adic valuation and the $ p $-adic absolute value? For sure I know how to do it by hand, but I want to check my results and rule out computational mistakes.

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WolframAlpha can do it. It understood my request 2-adic valuation of 42 as IntegerExponent[42,2], which seems to be the appropriate function for this. For the $p$-adic absolute value, there is the slight problem that there are several equivalent (as in: defining the same topology, not the same norm) choices. But replacing $n$ by $p^{-n}$ should not be too hard. (Plus, the valuation is more useful in practice).

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Maybe you can go to SageMathCloud. Create project, then create page and then use gp/pari on this page like this:

%gp
905/7 +O(7^3)

the answer will be

2*7^-1 + 3 + 4*7 + 2*7^2 + O(7^3)

You can see more in this video

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Late to the party here, but here's a very helpful Sage-implemented online base-n calculator which also does p-adics (check the box at the bottom):

https://billcookmath.com/sage/becimalCalculator.html

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  • $\begingroup$ Great tool, but unfortunately "while we allow p-adic expansions for arbitrary bases, p-adic expansion outputs require the output basis to be a prime number", which is pretty strange, since p-adic expansions exist for arbitrary bases, including composite bases as well. $\endgroup$ Mar 25 at 8:08

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