Title: Conjecture: the set of prime numbers is supernatural
Domain: Math

Question: | Formulate and analyze the conjecture that no non-constant natural function—built only from the variable $n$, integer constants, and the operations $+,\times,\wedge$—can generate a set of values consisting entirely of prime numbers. | Provide: | 1. A precise definition of natural functions and supernatural sets. | 2. A formal statement of the conjecture and its equivalence to the claim that the prime numbers form a supernatural set. | 3. Key implications, including consequences for Fermat numbers if the conjecture holds, and the extraordinary consequences if it fails. | |

Background: Fermat's Legacy and Natural Functions: | | 1. Historical Context: | Fermat's Conjecture: | - Original statement: {2^(2^n) + 1 | n ∈ Z⁺} ⊂ P | - Euler's disproval | - Impact on number theory development | | 2. Natural Functions Framework: | Definition: f: I → I where I = Z⁺ | Construction Elements: | - Variable n | - Constants from I | - Operations: +, ×, ^ | | Key Property: | Every natural function is constant or strictly increasing | Key Components: | a) Natural Numbers Abstraction: | - Fundamental counting principles | - Basic arithmetic operations | - Number theoretic properties | | b) Physical World Correlation: | - Geometric interpretations | - Measurement applications | - Real-world patterns |

Definitions: **Fermat’s Legacy and Natural Functions** | | 1. **Historical Context** | | * Fermat’s conjecture: numbers of the form $2^{2^n}+1$ are all prime. | * Euler disproved this by showing $2^{32}+1$ is composite. | * This highlights the difficulty of formulating formulas that only produce primes. | | 2. **Natural Functions Framework (Intuition)** | | * Defined on positive integers $I$. | * Built from the variable $n$, integer constants, and the operations $+,\times,\wedge$ (power). | * Key property: every natural function is either constant or strictly increasing. | * This motivates the question: can such “natural” constructions ever stay entirely within the primes? |

Methodology: | **Core Proposition and Consequences** | | 1. **Monotonicity of Natural Functions** | | * Proposition: every $f\in F_{\text{Nat}}$ is either constant or strictly increasing. | * Proof idea: induction on the construction length; use lemmas showing addition, multiplication, and exponentiation preserve strict increase. | | 2. **Conjecture Formulation** | | * For non-constant $f\in F_{\text{Nat}}$, $f(I)\not\subseteq P$. | * Equivalent to: $P$ is supernatural. | | 3. **Supporting Arguments** | | * Non-constant polynomials: periodicity mod $p$ ensures infinitely many composite values. | * Exponential forms $a^n+b$: Fermat’s little theorem forces composites. | * Fermat function: $F(5)$ composite. | | 4. **Implications** | | * If true: every natural function has infinitely many composite values; in particular, infinitely many composite Fermat numbers. | * If false: existence of an explicit arithmetic prime generator using only $+,\times,\wedge$.

Answer: | | ## 1. Basic objects and notation | | * Let $\mathbb N=\{0,1,2,\dots\}$, $\mathbb I=\{1,2,3,\dots\}$ (positive integers), and $\mathbb P=\{2,3,5,7,\dots\}$ (primes). | * Let $F=\{f:\mathbb I\to\mathbb I\}$. | | ### 1.1 Natural functions | | Define the **seed set** $F_s=\{\,n\mapsto n\,\}\cup\{\,n\mapsto a: a\in\mathbb I\,\}$. | Define three closure operators on subsets $P\subseteq F$: | | $$ | \begin{aligned} | A_+(P)&:=P\cup\{\,g+h: g,h\in P\,\},\\ | A_\times(P)&:=P\cup\{\,g\cdot h: g,h\in P\,\},\\ | A_\wedge(P)&:=P\cup\{\,g^\smallfrown h: g,h\in P\,\},\quad (g^\smallfrown h)(n):=g(n)^{\,h(n)}. | \end{aligned} | $$ | | Let $\Sigma$ be the set of all finite words over $\{A_+,A_\times,A_\wedge\}$. The class of **natural functions** is | | $$ | F_{\text{Nat}} \;:=\;\bigcup_{\sigma\in\Sigma}\sigma(F_s). | $$ | | Thus $f\in F_{\text{Nat}}$ iff $f$ can be built from $n$, integer constants, and the operations $+,\times,\wedge$ by finitely many compositions. Examples: every integer polynomial, $n\mapsto a^n+b$ with $a\in\mathbb I$, and the Fermat map $n\mapsto 2^{2^n}+1$. | | ### 1.2 Natural and supernatural sets | | An infinite $S\subseteq\mathbb I$ is **natural** if $S=f(\mathbb I)$ for some **non-constant** $f\in F_{\text{Nat}}$. | An infinite $S\subseteq\mathbb I$ is **supernatural** if it contains **no** natural subset, i.e., $\forall f\in F_{\text{Nat}}$ non-constant: $f(\mathbb I)\nsubseteq S$. | | --- | | ## 2. Fundamental monotonicity fact | | > **Claim (monotonicity).** Every $f\in F_{\text{Nat}}$ is either constant or strictly increasing on $\mathbb I$. | | *Proof sketch.* Define the **length** of $f$ as the minimal number of closure steps needed to generate $f$ from $F_s$. Induct on length. The base functions in $F_s$ are constant or strictly increasing. If $h,g$ are strictly increasing, then so are $h+g,\;h\cdot g,\;h^\smallfrown g$; if one is strictly increasing and the other constant, the same conclusions hold (except $1^\smallfrown h\equiv 1$). This closes the induction. | | Consequences: non-constant natural functions diverge and preserve order; in particular they cannot oscillate. | | --- | | ## 3. The conjecture and its equivalent reformulation | | ### 3.1 Formal conjecture | | > **Conjecture (no all-prime images for non-constant natural functions).** | > For every non-constant $f\in F_{\text{Nat}}$, $f(\mathbb I)\nsubseteq\mathbb P$. | | Equivalently, each such $f$ takes **some** composite value. | | ### 3.2 Equivalence to “the primes are supernatural” | | By definition of *supernatural*, the conjecture is exactly: | | $$ | \text{“\(\mathbb P\) is supernatural.”}\quad\Longleftrightarrow\quad | \forall f\in F_{\text{Nat}}\ \text{non-constant},\ f(\mathbb I)\nsubseteq\mathbb P. | $$ | | --- | | ## 4. Basic evidence and unconditional cases | | ### 4.1 Non-constant polynomials never stay prime | | Let $f\in\mathbb Z[n]$ be non-constant and map $\mathbb I\to\mathbb I$. Pick $n_0$ with $f(n_0)>1$ and a prime $p\mid f(n_0)$. Then $f(n_0+p)\equiv f(n_0)\equiv 0\pmod p$ but $f(n_0+p)>f(n_0)>1$, hence $f(n_0+p)$ is composite. Thus $f(\mathbb I)\nsubseteq\mathbb P$. | | ### 4.2 Exponential shifts $a^n+b$ | | Let $a>1,\ b\in\mathbb N$, $f(n)=a^n+b$. If $p\mid f(n_0)$ and $p\nmid a$, then by Fermat’s little theorem, | $a^{n_0+(p-1)}\equiv a^{n_0}\pmod p$, so $f(n_0+p-1)\equiv f(n_0)\equiv 0\pmod p$ and $f(n_0+p-1)>f(n_0)>1$, hence composite. If $p\mid a$ then $p<f(n_0)$ and $f(n_0)$ is already composite. So $f(\mathbb I)\nsubseteq\mathbb P$. | | ### 4.3 Fermat numbers | | For $F(n)=2^{2^n}+1$, one has $F(5)=4294967297=641\cdot 6700417$ (Euler), so $F(\mathbb I)\nsubseteq\mathbb P$. | | These three families are **natural functions**, so they match the conjecture’s prediction already in unconditional form. | | --- | | ## 5. Consequences conditional on the conjecture | | ### 5.1 Infinitely many composite values in each natural sequence | | For any non-constant $f\in F_{\text{Nat}}$, the set $\{x\in f(\mathbb I): x\ \text{composite}\}$ is infinite. | *Reason.* Since $f$ is strictly increasing, for each $k$ the shifted map $g(n)=f(n+k)$ is natural; by the conjecture $g$ attains a composite value at some $n=d$. Then $q=d+k>k$ and $f(q)$ is composite. As $k$ is arbitrary, we obtain infinitely many distinct composite values. | | ### 5.2 Infinitely many composite Fermat numbers | | Applying 5.1 to $F(n)=2^{2^n}+1$ shows there would be infinitely many composite Fermat numbers—precisely the long-standing open phenomenon this conjecture would settle. | | ### 5.3 If the conjecture were false | | A counterexample would be a **non-constant** $f\in F_{\text{Nat}}$ whose image consists **only** of primes. This would be an “arithmetical” closed-form prime generator using only $\{+,\,\times,\,\wedge\}$, producing arbitrarily large primes in order (since $f$ must be strictly increasing). Such a formula would be an extraordinary breakthrough and is widely considered implausible. | | --- | | ## 6. Supporting computational evidence (indicative) | | Systematic experiments on non-polynomial natural functions show that even when the first few values happen to be prime, a composite appears quickly. For example, besides the Fermat map, functions of the form $n\mapsto 2^{2^n}+2k+1$ were examined; several choices give primes for $n=1,\dots,4$, and one variant $2^{2^n}+93$ yields primes for $1\le n<7$, yet each sequence soon hits a composite value (see the tables and factorizations on pp. 5–6). These computations support the conjecture’s spirit that “simple” natural constructions cannot remain forever inside $\mathbb P$. &#x20; | | --- | | ## 7. Summary | | * **Definitions.** $F_{\text{Nat}}$ is the closure of $\{n\}\cup\{\text{constants}\}$ under $+,\times,\wedge$. A set is **supernatural** if it contains no image $f(\mathbb I)$ with $f\in F_{\text{Nat}}$ non-constant. | * **Fact.** Every $f\in F_{\text{Nat}}$ is constant or strictly increasing. | * **Conjecture (equivalent forms).** | (a) For every non-constant $f\in F_{\text{Nat}}$, $f(\mathbb I)\nsubseteq\mathbb P$. | (b) The prime set $\mathbb P$ is **supernatural**. | * **Unconditional cases.** Non-constant polynomials, $a^n+b$ with $a>1$, and the Fermat map already violate “all-prime image”. | * **Implications if true.** Each natural sequence has infinitely many composite values; in particular, **infinitely many composite Fermat numbers**. | * **If false.** Existence of a purely arithmetical, strictly increasing, closed-form prime generator—a result of extraordinary strength and unlikeliness. &#x20; | |

Checklist: 1. **Foundational Framework** | | * Define the sets of natural numbers $\mathbb N=\{0,1,2,\dots\}$, positive integers $\mathbb I=\{1,2,3,\dots\}$, primes $\mathbb P=\{2,3,5,7,\dots\}$, and the function space $F(\mathbb I,\mathbb I)$. | * Specify their basic set-theoretic properties with precise mathematical notation. | | 2. **Elevation Structure** | | * Define a structure $(E,+,\times,\wedge)$ with three binary operations. | * State and verify all axioms: commutativity, associativity, distributivity, and the elevation laws involving exponentiation. | * Show that these axioms are satisfied in multiple examples, such as integers and function spaces. | | 3. **Function Space Structure** | | * Prove that $F(\mathbb I,\mathbb I)$ with pointwise addition, multiplication, and exponentiation forms an elevation structure. | * Demonstrate closure under all operations. | | 4. **Morphisms** | | * Define evaluation maps $E_a: F(\mathbb I,\mathbb I)\to\mathbb I$ by $E_a(f)=f(a)$. | * Show that these maps preserve addition, multiplication, and exponentiation, hence qualify as morphisms of elevation structures. | | 5. **Operators and Construction Rules** | | * Define operators $A_+$, $A_\times$, and $A_\wedge$ on subsets of $F(\mathbb I,\mathbb I)$, each extending a set of functions by closure under one operation. | * Prove their properties, ensuring they can systematically generate new functions from given ones. | | 6. **Natural Functions** | | * Define the class of natural functions $F_{\text{Nat}}$ as the closure of constants and the identity function under repeated applications of $A_+$, $A_\times$, and $A_\wedge$. | * Show that $F_{\text{Nat}}$ itself forms an elevation structure. | * Provide examples, including polynomials, exponential-type maps, and Fermat-like functions. | | 7. **Monotonicity Property** | | * Prove that every natural function is either constant or strictly increasing. | * Use induction on construction length and establish the behavior of each operation when applied to constant or increasing functions. | | 8. **Supporting Lemmas** | | * Verify that addition, multiplication, and exponentiation preserve strict increase when applied to strictly increasing functions. | * Analyze mixed cases where one argument is constant and the other is strictly increasing. | | 9. **Special Cases of the Conjecture** | | * Demonstrate that certain families of natural functions cannot produce only primes: | | * Non-constant polynomials, | * Exponential forms such as $a^n+b$, | * Fermat numbers $2^{2^n}+1$. | | 10. **Natural and Supernatural Sets** | | * Define a natural set as the image $f(\mathbb I)$ of a non-constant natural function. | * Define a supernatural set as an infinite subset of $\mathbb I$ containing no natural set. | * State that the conjecture is equivalent to saying the set of prime numbers is supernatural. | | 11. **Consequences if the Conjecture Holds** | | * Show that every non-constant natural function must take infinitely many composite values. | * Deduce that infinitely many Fermat numbers are composite. | | 12. **Consequences if the Conjecture Fails** | | * Argue that there would exist an explicit, closed-form arithmetic formula using only $+,\times,\wedge$ that generates arbitrarily large prime numbers, a result of extraordinary significance. | | 13. **Technical Properties** | | * Define the length of a natural function as the minimal number of closure steps needed to construct it, and prove basic properties of length. | * Analyze growth behavior of natural functions, showing they are unbounded and strictly increasing when non-constant. | * Establish that the class is closed under composition, with preservation of growth and monotonicity. | |