Just a quick update, I’ll elaborate later. The technique I used 2 posts ago, in proving that infinitely often, can be quite easily generalised and yield a lot more that way. And it also gives some insight in why it’s hard to show infinitely often; maybe it’s not true! The theorem I will prove later (either today or tomorrow) will be:

Let be a given natural number and be a given prime. Let be the unique number smaller than such that for some . Then the following two are equivalent:

1)

2)

Where , with the least common multiple of

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