I spent a pleasant hour in a cafe in Oakland reconstructing the elementary proof of the “supplementary law” . This is exactly the kind of tricky elementary argument I have trouble remembering if I’ve only see it once before (which was the case). Anyway, the proof goes as follows: let . We’ll evaluate this mod *p* in two different ways. On the one hand, we clearly have . On the other hand, mod *p *we have .

Related puzzle: what is the value mod *p *of ? The first few values are misleading!

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