Thanks for the detailed response. I think I misunderstood your use of the pattern initially: you’re not saying that the pattern applies to 0 because that’s what leads to the pattern applying for 1 (which would be flawed logic as I described), you’re saying that 0 is one of the values in the pattern, for the first element in the pattern. Is that correct?

Let me just say that I agree that 0 is reasonably considered a natural number. As you pointed out, there are multiple definitions of natural numbers that are accepted by different mathematicians, or in different contexts, and at least one of those definitions consider 0 to be a natural number (e.g., the Cantor Set theory definition in which zero is defined as the empty set).

However, I‘m not yet convinced that your reasoning is valid. Is there some proof that all numbers appearing in this sequence must be natural numbers that doesn’t rely on first defining 0 as a natural number? Put another way, can you show that there is no alternative and meaningful definition of the natural numbers which does not include 0, but for which all of the other things you found to be true still hold? I suspect that there is, specifically I don’t think defining the natural numbers as all of the integers strictly greater than 0 will lead to any contradictions in what you’ve worked out.

In regards to the pattern: finding that it holds for all the primes in the range (3, 999999999999] is certainly compelling, but mathematics cannot rely on circumstantial evidence. Without a formal proof showing that it holds for any prime number greater than 3, it is only a conjecture, and not something that can be used to show that other statements are true.

At any rate, I applaud your curiosity and dedication. Kudos for all the great work you’ve put in!