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Here $\mathbb N = \{2,3,4,\dots\}$ with the binary operation of addition.

If $m \in \mathbb N$ we denote by $G_{\mathbb N} (m)$ the semigroup generated by $m$.

Definition: A number $p$ is said to be prime if for all $m \lt p$, $\;p \notin G_{\mathbb N} (m) $.

We denote the set of non-empty finite subsets of $\mathbb N$ by $\mathcal F (\mathbb N)$.

Let $\mathtt E$ be a function

$\quad \mathtt E: \mathbb N \to \mathcal F (\mathbb N)$

satisfying the following:

$\quad \quad\quad\forall n \in \mathbb N$

$\tag 0 \mathtt E (2) = \{2\}$

$\tag 1 \text{ If } (\forall \text{ prime } p \lt n) \; n \notin G_{\mathbb N} (p) \text{ then } \mathtt E (n) = \{n\}$

$\tag 2 \text{ If } \, (\exists \text{ prime } p \lt n) \; n \in G_{\mathbb N} (p) \text{ then } \mathtt E (n) \text{ is the union of all such primes}$

$\tag 3 \mathtt E (n+1) \cap \mathtt E (n) = \emptyset$

We have the following result:

Theorem 1: There exist one and only one function $\mathtt E$ satisfying $\text{(0)}$ thru $\text{(2)}$; it will also satisfy $\text{(3)}$. Moreover, for every $n$, all the numbers in the set $\mathtt E (n)$ are prime (the prime 'factors').

Question: Can the theorem be proved in this $(\mathbb N,+)$ setting?

If yes, we can continue.

Theorem 2: The set of all prime numbers is an infinite set.
Proof
If $a_1$ is any number, consider the 'next further out' number

$\tag 4 a_2 = \sum_{i=1}^{a_1+1}\, a_1 = \sum_{i=1}^{a_1}\,( a_1 + 1)$.

A simple argument using $\text{(3)}$ shows that $\mathtt E (a_1) \subsetneq \mathtt E (a_2)\;$ (c.f. Bill Dubuque's remark).

Employing recursion we get a sequence $a_1, a_2, a_3,\dots$ with a corresponding chain of strictly increasing sets

$\quad \mathtt E (a_1) \subsetneq \mathtt E (a_2) \subsetneq E (a_3) \dots$

So there are sets of prime numbers with more elements than any finite set. $\blacksquare$

My Work

Please see

Using the recursion theorem to implement the Sieve of Eratosthenes.

The proof of theorem 2 is along the lines found in the proof given by Filip Saidak. Also, if we set $a_1$ to $1$ in theorem 2 we get the researched OEIS sequence A007018.

Note that the proof supplied by Filip Saidak has most likely been known for many years; see Bill Dubuque's answer to the math.stackexchange.com question

Is there an intuitionist (i.e., constructive) proof of the infinitude of primes?

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  • $\begingroup$ Note, multiplication (of positive integers) is recursive addition. That is, we define $a\cdot1$ to be $a$ and $a\cdot(n+1)$ to be $a\cdot n+a$. $\endgroup$ – Barry Cipra Nov 22 '18 at 17:39
  • $\begingroup$ @BarryCipra Yes I know. But I get excited when something that appears 'bound to multiplication' - the prime numbers - can get released into a more elementary framework. $\endgroup$ – CopyPasteIt Nov 22 '18 at 17:44
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    $\begingroup$ The proof via $\,n(n\!+\!1)\,$ has more prime factors than $n$ for $\,n\ge 1$ is surely much older than Saidak's 2005 paper. The generated sequence is OEIS A007018. Note: adding $1$ yields Sylvestoer sequence $a_{n+1} = a_n^2 - a_n + 1 = $ OEIS A000058. See the OEIS notes for other connections. $\endgroup$ – Bill Dubuque Nov 22 '18 at 18:05
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    $\begingroup$ @CopyPasteIt The least factor $> 1$ of $n$ is prime (i.e. irreducible) is already quite minimal. $\endgroup$ – Bill Dubuque Nov 22 '18 at 19:08
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    $\begingroup$ @CopyPasteIt Well one can unwind everything down to Peano arithmetic, but that won't be very arithmetically enlightening. Why prefer assembly language over the beautiful higher-level language carefully crafted by number theorists over many centuries? $\endgroup$ – Bill Dubuque Nov 22 '18 at 19:37
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It is valid, but seems to be fundamentally the same as the classical proof; both hinge upon the fact that $p_1\times p_2\times \cdots\times p_n+1$ is divisible by some prime not in $\{p_1,p_2,\dots,p_n\}$.

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    $\begingroup$ @Copy It is poor form to change the question in a way that invalidates existing answers. $\endgroup$ – Tobias Kildetoft Nov 23 '18 at 17:08

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