slug: ayad-bouchenna-base-b-reverse-multiple-divisors bibkey: ayad2025reversemultiples doi: 10.5281/zenodo.15283699 url: https://math.colgate.edu/~integers/z37/z37.pdf triage: theorem motivation_gids:
- D5/S1/Digit/AyadBouchennaReverseMultipleDivisors.result
Problem
Mohamed Ayad and Rachid Bouchenna, Which Integer Divides the Reverse of Any of Its Multiples?, INTEGERS 25 (2025), #A37, printed p. 9, Problem 1:
Let B ≥ 2 be any number base. Are the divisors of B² − 1 the only positive integers satisfying the property P*_B ?
Section 4.1 on printed p. 8 defines the base-B reverse m*_B by reversing
the positional digits of the positive integer m. The property P*_B means
that n divides m*_B for every positive multiple m of n. The exact statement
is: for all natural B ≥ 2 and all natural n > 0,
(∀ m : ℕ, 0 < m → n ∣ m → n ∣ Nat.ofDigits B (Nat.digits B m).reverse) ↔ n ∣ B ^ 2 - 1.
Motivation
The paper’s Theorem 3 characterizes the base-ten case by the divisors of 99. Proposition 5 on printed p. 8 proves sufficiency for every base. The question asks whether necessity also holds uniformly in the base. The motivation GID supplies this full characterization, including both directions.
Gap
The arbitrary-base necessity is the question absent from Proposition 5. Preregistration #8990 records the fully quantified statement, tier-one classification and literature readings before the probe. The orchestrator’s September 20, 2026 readings are: arXiv exact phrase and author queries each returned zero entries; Crossref had no matching index entry; MathDB had no entry for this question, with a Collatz positive control returning 20; OpenAlex indexed the article with cited_by_count zero. OEIS A018282 covers only the already established base-ten case. These searches bound the literature claim to their queried scope.
Route
A positive multiple whose leading base-B digit is one forces gcd(n,B)=1
whenever n has the reversal property. For a unit B modulo n, let T be its
multiplicative order. If B² is not one modulo n, then T ≥ 2. Sparse digit
lists with ones at positions 0, 1, T, …, (c+1)T have forward residue
1+B+(c+1) and B times their reversed residue equal to 1+B+(c+1)B.
Choosing c congruent to −(2+B) makes the forward residue zero. The reversal
property then forces B²=1, a contradiction. This is the private witness
lemma used in result. Conversely, B²=1 gives the digit-reversal identity
by induction and preserves divisibility for every multiple.
Falsifier
A base B ≥ 2 and positive n with the reversal property but n not dividing B²−1 would refute necessity. A positive divisor n of B²−1 and a positive multiple m with n not dividing m*_B would refute sufficiency. The theorem ranges over all such B, n and m; bounded enumerations alone are insufficient.
Evidence
The Lean source is D5/S1/Digit/AyadBouchennaReverseMultipleDivisors.lean.
Its public definitions are reverseBase and HasReverseMultipleProperty;
its single public theorem is result. The Scribe theorem node links this
question to that result with ResolutionKind.Proved. Reversal uses the
canonical Mathlib digits and handles trailing zeros by positional evaluation.
The orchestrator independently computed the property from its definition for bases 2 through 16, n ≤ 400 and every multiple t ≤ 4000, obtaining the divisors of B²−1 with zero extras and zero misses. At B=10 the set is {1,3,9,11,33,99}, matching Theorem 3. These are attributed bounded readings, not the justification for the universal quantifiers.
Triage
theorem: the arbitrary-base characterization is an unbounded symbolic
result. No restriction to a finite range of bases, moduli or multiples is
part of the conclusion. Problem 2 and the weaker property in section 4.2
are outside this statement.
ASSUMED-UNVERIFIED
The literature and numerical readings above are supplied orchestrator readings; this Stage B seat did not repeat those external computations or literature queries. The source clauses and preregistered statement were checked against issue #8990 and the supplied Library note. Exhaustive publication priority outside the searched scope is unverified. Semantic fidelity and proof-shape classification remain review judgments rather than consequences of the resolution marker validator.