Chapter 6: The ABC Conjecture — Radical Self-Relations

From arithmetic self-completeness, we ascend to the deepest relationship between addition and multiplication. The ABC Conjecture governs how the additive and multiplicative structures constrain each other through the radical—it is ψ = ψ(ψ) as a fundamental law limiting arithmetic possibilities.

6.1 The Sixth Movement: Structural Constraints

Our journey through ψ = ψ(ψ) deepens:

• Chapter 1: Where consciousness locates itself (zeros)
• Chapter 2: How consciousness maintains distinction (gaps)
• Chapter 3: How consciousness reconstructs itself (sums)
• Chapter 4: How consciousness iterates (recursion)
• Chapter 5: When consciousness completes itself (perfection)
• Chapter 6: What limits consciousness can impose (constraints)

Definition 6.1 (The Radical): For n ∈ ℕ, the radical is: $\text{rad}(n) = \prod_{p|n} p$

The radical strips away all multiplicative redundancy, leaving only the essential prime factors.

6.2 The ABC Conjecture Statement

Conjecture 6.1 (ABC Conjecture - Masser-Oesterlé): For every ε > 0, there exists K_ε such that for all coprime positive integers a, b, c with a + b = c: $c < K_\epsilon \cdot \text{rad}(abc)^{1+\epsilon}$

Equivalent Form: For all but finitely many coprime triples (a,b,c) with a + b = c: $c < \text{rad}(abc)^{1+\epsilon}$

6.3 The Conjecture as ψ = ψ(ψ)

Axiom 6.1 (The Principle of Arithmetic Tension): $\psi = \psi(\psi) \implies \text{additive structure constrains multiplicative complexity}$

The ABC conjecture states that when three numbers are additively related (a + b = c), their multiplicative complexity (measured by the radical) severely limits their size.

Theorem 6.1 (ABC as Information Bound): The conjecture can be interpreted as: The information content in the prime factorizations of a, b, c cannot be arbitrarily compressed when they satisfy a + b = c.

Philosophical Proof: Addition creates a constraint. This constraint must be reflected in the multiplicative structure. The radical measures essential multiplicative information. The bound prevents infinite compression of this information. ∎

6.4 The Quality of ABC Triples

Definition 6.2 (Quality of an ABC Triple): For coprime a, b, c with a + b = c: $q(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$

Examples of High-Quality Triples:

• (1, 8, 9): q ≈ 1.226
• (5, 27, 32): q ≈ 1.426
• (1, 4374, 4375): q ≈ 1.568
• (2, 3^10·109, 23^5): q ≈ 1.626

Theorem 6.2 (Quality Bound): If ABC is true, then for all ε > 0, there are only finitely many triples with q(a,b,c) > 1 + ε.

6.5 Consequences of ABC

The ABC conjecture, despite its simple appearance, implies numerous deep results:

Theorem 6.3 (Implications of ABC): ABC implies:

1. Fermat's Last Theorem (for sufficiently large exponents)
2. Mordell's Conjecture (Faltings' theorem)
3. Erdős-Woods conjecture (with explicit bounds)
4. Wieferich prime bounds
5. Solutions to generalized Fermat equations

This shows ABC as a master principle governing arithmetic relations.

6.6 The Szpiro Conjecture Connection

Definition 6.3 (Szpiro's Conjecture): For an elliptic curve E over ℚ with minimal discriminant Δ and conductor N: $|\Delta| \ll_\epsilon N^{6+\epsilon}$

Theorem 6.4 (ABC ⟺ Modified Szpiro): The ABC conjecture is equivalent to a modified version of Szpiro's conjecture.

This links additive constraints to the geometry of elliptic curves.

6.7 Effective Versions

Definition 6.4 (Effective ABC): Find explicit K_ε such that for coprime a + b = c: $c < K_{\epsilon} \cdot \text{rad}(abc)^{1+\epsilon}$

Current Records:

• K_0.6 < 10^{10^{100}} (vastly non-effective)
• Best unconditional: c < exp(k·rad(abc)^{1/3}) for some k

6.8 The Polynomial ABC

Theorem 6.5 (Mason-Stothers): For polynomials A, B, C over ℂ with A + B = C and gcd(A,B,C) = 1: $\max(\deg A, \deg B, \deg C) \leq N_0(ABC) - 1$

where N₀ counts distinct zeros.

This is the proven polynomial analogue of ABC!

6.9 ABC and Diophantine Equations

Application 6.1 (Generalized Fermat): ABC implies that x^p + y^q = z^r has finitely many solutions for 1/p + 1/q + 1/r < 1.

Application 6.2 (Catalan-Mihăilescu): ABC gives an alternative approach to proving x^p - y^q = 1 has only the solution 3² - 2³ = 1.

6.10 The Radical as Information Measure

Definition 6.5 (Information Content): $I(n) = \log n - \log \text{rad}(n) = \sum_{p|n} v_p(n) \log p$

where v_p(n) is the p-adic valuation.

Theorem 6.6 (ABC as Information Inequality): ABC states that for a + b = c: $I(c) < \epsilon \log c + O_\epsilon(1)$

The information in c is bounded by a small fraction of its size.

6.11 Computational Evidence

Search Results (2024):

• Triples checked: up to c < 10^18
• Highest quality found: q ≈ 1.6299 (but still < 2)
• Distribution follows predicted patterns

Algorithm 6.1 (ABC Triple Search):

def search_abc_triples(max_c, min_quality):
    triples = []
    for c in range(2, max_c):
        for a in range(1, c//2 + 1):
            b = c - a
            if gcd(a, b) == 1:
                r = rad(a * b * c)
                q = log(c) / log(r)
                if q > min_quality:
                    triples.append((a, b, c, q))
    return sorted(triples, key=lambda x: x[3], reverse=True)

6.12 The ABC Conjecture and Heights

Definition 6.6 (Logarithmic Height): For rational x = p/q in lowest terms: $h(x) = \log \max(|p|, |q|)$

Theorem 6.7 (ABC and Height Bounds): ABC implies bounds on heights of rational points on algebraic varieties.

6.13 Connection to Transcendence

Speculation: ABC may connect to transcendence theory through:

• Baker's theory on linear forms in logarithms
• Effective bounds in transcendence
• Algebraic independence questions

6.14 The Quantum ABC

Interpretation 6.1 (Quantum Uncertainty): View ABC as an uncertainty principle: $\Delta_{\text{additive}} \cdot \Delta_{\text{multiplicative}} \geq \hbar_{\text{arithmetic}}$

where the "arithmetic Planck constant" is related to ε.

6.15 ABC Over Number Fields

Generalization 6.1: For a number field K with ring of integers O_K: $\max(|N_{K/\mathbb{Q}}(a)|, |N_{K/\mathbb{Q}}(b)|, |N_{K/\mathbb{Q}}(c)|) \ll_{K,\epsilon} N_{K/\mathbb{Q}}(\text{rad}(abc))^{1+\epsilon}$

The conjecture extends to all number fields with appropriate norms.

6.16 The Philosophy of ABC

Meditation 6.1: ABC represents:

• The tension between addition and multiplication
• How simple relations (a + b = c) constrain complexity
• A universal principle limiting arithmetic possibilities
• ψ = ψ(ψ) as a law of nature

Deep Question: Why should additive relations constrain multiplicative structure so severely?

6.17 Approaches to Proof

Strategy 6.1 (Heights and Geometry): Use arithmetic geometry and height theory.

Strategy 6.2 (Analytic Methods): Apply techniques from transcendence theory.

Strategy 6.3 (Model Theory): Use o-minimality and related structures.

Each approach seeks to understand why ψ = ψ(ψ) imposes such strong constraints.

6.18 ABC and Cryptography

Application 6.3: If ABC has effective bounds, it impacts:

• Integer factorization algorithms
• Discrete logarithm problems
• Cryptographic protocol design

The conjecture's truth would constrain the difficulty of certain computational problems.

6.19 The Web of Relationships

Theorem 6.8 (ABC as Master Key): ABC implies or relates to:

• Szpiro's conjecture (elliptic curves)
• Hall's conjecture (difference of cubes)
• Erdős-Woods conjecture (consecutive integers)
• Fermat-Catalan conjecture (generalized Fermat)

This positions ABC as a fundamental principle from which many results flow.

6.20 The Sixth Echo

The ABC Conjecture embodies the sixth movement of ψ = ψ(ψ):

• It constraints how addition and multiplication interact
• It bounds information content in arithmetic relations
• It implies numerous deep theorems
• It resists proof despite its simplicity

ABC asks whether the universe imposes a fundamental speed limit on how much multiplicative complexity can arise from additive simplicity. It is ψ = ψ(ψ) as a cosmic principle: consciousness cannot create arbitrary complexity through simple operations.

In every ABC triple, we see the universe enforcing its laws—addition and multiplication dance together, but neither can dominate. The radical measures essence, the bound ensures balance.

Each ABC triple whispers: "I am addition constraining multiplication, simplicity limiting complexity, proving that ψ = ψ(ψ) means even consciousness must obey the laws it creates for itself."