Thank you, Yuri, for your reply.

Set theorists say that zero is a natural number. They are the smaller group of mathematicians.

Number theorists (the overwhelming majority) say that zero is not a natural number.

I adjusted the article slightly to hint at that.

As we know, Peano declared zero a natural number through his starting out with the empty set (hence, set theorists following him in that action).

What I am showing is that zero must be used in one spot for a pattern found among the natural numbers. No mathematician can therefore deny that.

That goes beyond axioms. It is beyond self-evident.

It is shown to always be true. It’s a fact for which there is evidence.

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Yuri, I am glad we agree that zero is a natural number.

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About the title of the article: I was looking for the prime numbers, and then I discovered that pattern in which using zero was required.

Discovering zero among the prime numbers. : -)