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"Elementary" is a somewhat subjective term that roughly means "from first principles", i.e. not relying on a bunch of advanced theorems and concepts. It doesn't necessarily mean the explanation is simple or concise.

Often the advanced theorems and concepts make for a simpler and more concise explanation.

I guess the closest equivalent in programming is "implement X without using any external libraries (or perhaps only the standard library)".

Elementary number theory means that you are doing number theory without complex analysis or lots of abstract algebra

Elementary means not using complex numbers.

“Elementary” is a relative term!