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E.g., can I write $(a^{p})^{2p} \equiv a^{2p}=a^pa^p\equiv aa\equiv a^2\pmod{\! p}$?

I often see equality symbols inbetween mod equivalences. The equality signs point out the equality is not restricted to mod $p$ and holds in general.

Can it technically be used this way? I know it is not ambiguous, it is understandable and may make some things clearer, and I know this question is nitpicking about notation. But I'm thinking maybe there is some consensus regarding this?

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What you wrote is correct. You can also use everywhere equivalence signs, but not everywhere equality signs. What you wrote is the best.

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Better to write all the symbols in the equivalent form, because if you use equivalent even once, then at the end you can just deduce a equivalent relation not equality, so because every equality implies equivalent in every modulo better to write all the symbols in equivalent mode.

Your note is clear enough, but is not symmetric...using alternative symbols makes your note less neat...

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    $\begingroup$ I disagree. For example, it is common to use chains of equality and less-than signs. Indicating actual equalities helps because it decreases the number of possibilities I have to consider when trying to understand the argument. The same applies here. $\endgroup$ – Eike Schulte May 17 '15 at 18:21

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