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Since I've produced many ramblesbelow, I thought I'd close my (anti-)contribution with a distilled version of the example I've attempted below. It's still something very standard, but, I hope, in the spirit of the original question. I'll describe it as if it were a personal thing.

It's quite a compelling to view image, I think, an integer as a function that varies in this way for a while before eventually settling downleveling off. But, for various a number of reasons, I rarely say this mention it to students or even to colleagues. Maybe I should.

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Since I've produced many rambles below, I thought I'd close my (anti-)contribution with a distilled version of the example I've attempted below. It's still something very standard, but, I hope, in the spirit of the original question. I'll describe it as if it were a personal thing.

Almost always, I think of an integer as a function of the primes. So for 20, say,

20(2)= 0

20(3)=2

20(5)=0

20(7)=6

20(19)=1

20(23)=20

20(29)=20

20(31)=20

20(37)=20

It's quite compelling to view an integer as a function that varies in this way for a while before eventually settling down. But, for various reasons, I rarely say this to students or even to colleagues. Maybe I should.

It's unclear if this is an appropriate kind of answer, in that I'm not putting forward anything very specific.

By the way, my choice of example was actually somewhat motivated by the fact that it is quite likely to be difficult for people outside of arithmetic geometry, including many readers of this forum. This gives it a different flavor from the situations where we all understand each other more or less well, and focus therefore on pedagogical issues referring to classroom practice.

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Unfortunately, this does involves involve some mention of sheaves, arithmetic fundamental groups, and some other strange things. Afterwards,

As to $Spec(\Bbb{Z})$, perhaps there will be occasion to bore you with that some other time. Why don't you post a question (assuming you are interested)? Then you are likely to get a great many perspectives more competent than mine. It might be an interesting experiment relevant to your original question.