[Paper Review] A Proof of the Loehr-Warrington Amazing TEN to the Power n Conjecture

Shalosh B. Ekhad, Vince Vatter+1 more|ArXiv.org|Sep 15, 2005

semigroups and automata theory3 citations

TL;DR

This paper proves the Loehr-Warrington conjecture that there are exactly $10^n$ words of length $5n$ over the alphabet \{3, -2\} that sum to zero and avoid the factor $3[-2]$ where the inner word sums to zero. The proof uses a computer-assisted method: a custom Maple package called TEN automatically discovers a linear grammar for the language, rigorously verifies its correctness, and computes the generating function $1/(1 - 10x^5)$, confirming the $10^n$ count via automated symbolic computation.

ABSTRACT

We prove, via 30 seconds of Maple computation, that there are 10^n words in the alphabet {3,-2} of length 5n, sum 0, and such that every factor that sums to 0 and that starts with a 3 may not be immediately followed by a -2.

Motivation & Objective

• To prove the conjecture that there are $10^n$ words of length $5n$ over \{3, -2\} with sum zero and no forbidden factor $3[-2]$ where the inner word sums to zero.
• To develop and apply an automated method for discovering and verifying linear grammars for combinatorial languages with sum and factor constraints.
• To demonstrate that computer-assisted proof systems can handle nontrivial combinatorial conjectures that are difficult to resolve by hand.
• To provide a publicly available Maple package (TEN) that automates the discovery and verification process for similar problems.

Proposed method

• The Maple package TEN generates all valid words up to a given length and constructs a binary family tree of word pairs $[w_1, w_2]$ representing sublanguages $L(w_1, w_2)$.
• The algorithm uses two partitioning rules—HeadWay and TailWay—based on the first and last letters of the middle word, recursively splitting sublanguages into sub-sublanguages.
• It detects and eliminates empty or cloned sublanguages (isomorphic to earlier ones) to prune the search space and build a minimal grammar.
• The grammar is verified for correctness using a recursive proof procedure based on the Discrete Rolle Theorem, which decomposes words by their minimal positive or negative prefix sums.
• The method applies purges to eliminate any meta-words containing zero-sum proper factors or forbidden mishaps (e.g., $a[-a](-b)$), ensuring only valid factorizations remain.
• The final generating function is computed as $GF = 1/(1 - 10x^5)$, confirming the $10^n$ count for words of length $5n$.

Experimental results

Research questions

• RQ1 Are there exactly $10^n$ words of length $5n$ over the alphabet \{3, -2\} that sum to zero and avoid the factor $3[-2]$ where the inner word sums to zero?
• RQ2 Can a computer-assisted system automatically discover and rigorously verify a linear grammar for a complex combinatorial language with sum and factor constraints?
• RQ3 What is the structure of the language of zero-sum words avoiding specific forbidden subfactors, and can it be captured by a regular grammar?
• RQ4 Can the method used for $a=3, b=2$ be generalized to other relatively prime integers $a, -b$?
• RQ5 What is the role of automated proof systems in replacing or augmenting human intuition in combinatorial mathematics?

Key findings

• The number of valid words of length $5n$ over \{3, -2\} that sum to zero and avoid the forbidden factor $3[-2]$ is exactly $10^n$, confirming the Loehr-Warrington conjecture.
• The Maple package TEN successfully discovered a linear grammar for the language, which was then rigorously verified as correct using automated proof techniques.
• The generating function for the language is $1/(1 - 10x^5)$, which directly implies the $10^n$ count for words of length $5n$.
• The proof method relies on recursive decomposition via the Discrete Rolle Theorem, which partitions words by their minimal positive or negative prefix sums.
• Three purges were sufficient to eliminate all invalid meta-words (containing zero-sum factors or mishaps), proving that no minimal counterexamples exist.
• The method was successfully applied to the $\{3, -2\}$ case, and the generalization to $b=2$ and odd $a$ was later proven by human mathematicians inspired by this automated proof.