On the Extended 1-2-3 Conjecture of Pilz

Abstract: We resolve (for all sufficiently large $n$) a conjecture of Pilz on the symmetric difference $AΔ(2A)Δ\cdotsΔ(nA)$ for finite sets $A\subseteq \mathbb{N}$ of positive integers. We show that this set always has cardinality at least $n$ for large $n$.

• The paper resolves Pilz’s Extended 1-2-3 Conjecture for all sufficiently large n using multiscale induction and combinatorial techniques.
• It establishes explicit lower bounds for the symmetric difference of set operations by leveraging algebraic frameworks and p-adic slicing methods.
• The work bridges additive combinatorics and analytic number theory, offering actionable methods for computing bounds and analyzing set structures.

Resolution of the Extended 1-2-3 Conjecture of Pilz

Introduction and Context

The paper resolves a longstanding open problem in additive and combinatorial number theory, specifically the Extended 1-2-3 Conjecture posed by Pilz. Let $A$ be a finite subset of $\mathbb{N}$, and for any $n \in \mathbb{N}$ consider the set

$$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$

where $\ast$ denotes a specific binary operation derived from the parity of product representations and $\Delta$ is the symmetric difference. Pilz conjectured that $|A \ast [n]| \ge n$ for all finite, nonempty $A \subseteq \mathbb{N}$ and all $n$. This "Extended 1-2-3 Conjecture" is motivated by notions in coding theory related to linear codes' minimal distances and parity questions for product representations.

Prior to this work, the result was confirmed for small $n$, the special case $\mathbb{N}$0 ($\mathbb{N}$1 equals the first $\mathbb{N}$2 positive integers), and certain restricted $\mathbb{N}$3, with only sublinear lower bounds known in general. The present paper fully resolves Pilz's conjecture for all sufficiently large $\mathbb{N}$4.

Algebraic Framework and Set Operations

The authors develop a formal algebraic framework for manipulating $\mathbb{N}$5 and $\mathbb{N}$6 on finite subsets of $\mathbb{N}$7. They show that $\mathbb{N}$8 forms a commutative ring of characteristic $\mathbb{N}$9, with explicit ring isomorphism to $n \in \mathbb{N}$0, where monomials correspond to products of distinct primes. This translation enables the application of combinatorial and algebraic methods to analyze the problem.

A key technical tool is the inclusion-exclusion formula for $n \in \mathbb{N}$1, allowing precise calculation and the development of effective lower bounds. The symmetric difference captures elements with odd numbers of product representations, and product-slicing via $n \in \mathbb{N}$2-adic valuation and smooth-rough decompositions is foundational.

Resolution for Small $n \in \mathbb{N}$3 and Special Cases

The authors provide a complete proof of the conjecture for all $n \in \mathbb{N}$4 and comprehensively analyze the $n \in \mathbb{N}$5 cases using $n \in \mathbb{N}$6-adic slicing strategies (splitting $n \in \mathbb{N}$7 according to divisibility by $n \in \mathbb{N}$8) and combinatorial manipulations, confirming that $n \in \mathbb{N}$9 always holds in these cases. For $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$0 small compared to $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$1, asymptotic lower bounds and explicit estimates are derived, improving earlier results and yielding accurate $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$2 for small $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$3 via computational and structural arguments.

Lower Bounds and Structure for General $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$4

For general sets $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$5, a recursive and combinatorial argument gives significant progress. The paper introduces:

• Partitioning and Slicing: Given $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$6 (disjoint), explicit lower bounds for $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$7 in terms of $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$8, $$A \ast [n] = A \Delta (2A) \Delta \cdots \Delta (nA),$$9 and the largest prime divisor of elements in $\ast$0 are established.
• Smoothness vs. Roughness: For "smooth" $\ast$1 (where all elements have small prime factors), analytic number theory (notably bounds for rough numbers in intervals) is used to give asymptotically sharp lower bounds.
• Algebraic and Polynomial Techniques: When $\ast$2 contains elements with large distinct prime divisors, the partitioning technique is highly effective, and the algebraic framework allows reduction to a finite computational problem in small cases.

The authors provide a method for systematically computing $\ast$3 (and thus minimal asymptotic ratios for fixed $\ast$4) and prove explicit uniform lower bounds, culminating in $\ast$5 for all but the degenerate cases $\ast$6 or $\ast$7.

Main Theorem for Large $\ast$8

The central result is that for all $\ast$9 (with explicit though very large $\Delta$0), for every finite, nonempty $\Delta$1,

$\Delta$2

thus confirming the conjecture. The proof is technically involved and relies on multiscale induction—removing primes from a large interval and careful control over the combinatorics of prime factorizations. Key arguments include:

• p-adic Slicing: Iteratively splitting $\Delta$3 by their divisibility by large primes in $\Delta$4 and reducing the problem to bounding set expressions after removing exceptional primes.
• Power Slicing and Symmetric Difference Chains: Handling the structure of sets $\Delta$5 where only small number (often powers of $\Delta$6) occur in the exponents, allowing the use of combinatorics of set chains and inclusion-exclusion.
• Intersection Estimates: Controlling the intersection size between distinct such sliced sets, leveraging properties of high-order differences and powers, and utilizing explicit estimates for the distribution of primes (e.g., from Rosser and Schoenfeld).
• Final Inductive Step: If at any step the sum of chain cardinalities exceeds a small threshold, the total is greater than $\Delta$7; otherwise, the residual $\Delta$8 is so small that the result follows from explicit computation or previously established bounds.

The explicit constant $\Delta$9 is truly enormous, but improvements would require sharper analytic number theory or more combinatorial innovation.

Implications, Contradictory/Strong Claims, and Related Problems

• Uniform Bound: The paper establishes that the symmetric difference in the conjecture never becomes anomalously small, unifying and extending all previously known special cases.
• Structural Limitation: The result crucially uses the arithmetic structure of $|A \ast [n]| \ge n$0; the conjecture does not hold (even asymptotically) for arbitrary subsets $|A \ast [n]| \ge n$1, as explicit counterexamples are provided.
• Sharpness and Limiting Cases: For certain $|A \ast [n]| \ge n$2, the lower bound is tight (equality is attained for $|A \ast [n]| \ge n$3 and $|A \ast [n]| \ge n$4).
• Connections: The work links to additive combinatorics (parity and cancellation in products) and combinatorial geometry (cf. a related open question on the symmetric difference of discs).
• Algorithmic Consequences: The explicit bounds and methods enable effective computation of $|A \ast [n]| \ge n$5 for moderate $|A \ast [n]| \ge n$6.
• Open Directions: For smaller $|A \ast [n]| \ge n$7, better than gigantic constants for $|A \ast [n]| \ge n$8 are possible, but may require fundamentally new ideas, particularly for the lower range.

Conclusion

This paper rigorously settles the Extended 1-2-3 Conjecture of Pilz for all sufficiently large $|A \ast [n]| \ge n$9, using sophisticated combinatorial, algebraic, and analytic techniques. The transition from lower bounds of order $A \subseteq \mathbb{N}$0 to uniform linear lower bounds represents a major theoretical advance. The analytic machinery developed provides new avenues in the interaction of combinatorics, algebra, and analytic number theory, and sets the stage for further exploration of arithmetic set operations and their extremal properties.

Reference: "On the Extended 1-2-3 Conjecture of Pilz" (2607.00934)