Let K_n and F_n denote the Kurepa factorial sequences, and let a be a constant. Define gcd(K_n+a,K_n+1+a)=gcd(F_n+a,F_n+1+a)=F_n(a). Bounded-gcd conjecture. For every Kurepa factorial sequence, F_n(a) is bounded above by 2, with F_n(a)= F_n(0), & n>1, F_n(4), & n≥ 1, where the values F_n(0) and F_n(4) are bounded above by 2; the question is to determine all values of a for which this bound holds. The supplied source does not establish the conjecture or resolve the accompanying classification question.

Bounded-gcd conjecture for shifted Kurepa factorial sequences

About 1 year old · traced to 2025

Let $K_n$ and $\mathbb{F}_n$ denote the Kurepa factorial sequences, and let $a$ be a constant. Define

\gcd(K_n+a,K_{n+1}+a)=\gcd(\mathbb{F}_n+a,\mathbb{F}_{n+1}+a)=\mathcal{F}_n(a).

Bounded-gcd conjecture. For every Kurepa factorial sequence, $\mathcal{F}_n(a)$ is bounded above by $2$ , with

\mathcal{F}_n(a)= \begin{cases} \mathcal{F}_n(0), & n>1,\\ \mathcal{F}_n(4), & n\geq 1, \end{cases}

where the values $\mathcal{F}_n(0)$ and $\mathcal{F}_n(4)$ are bounded above by $2$ ; the question is to determine all values of $a$ for which this bound holds. The supplied source does not establish the conjecture or resolve the accompanying classification question.

References

Primary source

Francis Atta Howard, “Partition Functions and Kurepa Decomposition I: Algebraic computation and some physical Applications”, arXiv:2509.06077 (2025).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

The original conjecture is challenged by an unverified counterexample, and an accompanying unverified argument claims to classify all integer shifts that could satisfy the bound.

The September 2025 paper formulates the shifted-gcd conjecture and asks which constants $a$ give a universal bound of $2$ ; it does not claim a classification or resolution.

Known results

The unshifted Kurepa case is identified with the open assertion $\gcd(K_n,n!)=2$ for $n>1$ (Kurepa, 1971).
The paper records $\mathcal{F}_n(4)=1$ at $n=0$ and $2$ otherwise, but presents the broader shifted claim as Conjecture 2.
It also gives explicit exceptional values for $a=2$ , including $\mathcal{F}_6(2)=6$ and $\mathcal{F}_n(2)=12$ otherwise.

Posted attempt

An unverified reader-written calculation claims $\gcd(K_{17}+4,K_{18}+4)=34$ and, in fact, divisibility by $34$ for every $n\geq17$ , refuting the proposed shift $a=4$ . It further claims a complete criterion: the bound holds for all $n\geq2$ exactly when $a\not\equiv2\pmod4$ and $a\not\equiv-K_p\pmod p$ for every odd prime $p$ . This attempt has not been independently verified.

Current status (as of August 2026): The paper's conjecture remains unproved, while a reader-written counterexample and purported classification claim would settle it negatively if verified.

Sources

CounterexampleThis solution needs a summarySee full solutionHide full solution

Counterexample for the conjectured shift $a=4$ , with infinitely many failures.

Let

K_n={!n}=\sum_{j=0}^{n-1}j!, \qquad K_{n+1}-K_n=n!.

The source's other sequence is $F_n=K_{n+1}$ , so its indices are shifted by one.

Direct exact evaluation gives

K_{17}=22\,324\,392\,524\,314\equiv13=-4\pmod{17},

and

K_{18}=378\,011\,820\,620\,314.

Therefore

\boxed{\gcd(K_{17}+4,K_{18}+4) =\gcd(K_{17}+4,17!)=34>2.}

Moreover, for every $n\ge17$ ,

K_n-K_{17}=\sum_{j=17}^{n-1}j!\equiv0\pmod{17}.

Hence $17\mid K_n+4$ for every $n\ge17$ . Also $K_n$ is even for every $n\ge2$ . Consequently

\boxed{34\mid\gcd(K_n+4,K_{n+1}+4) \qquad\text{for every }n\ge17.}

Equivalently, with the source's $F$ -indexing,

34\mid\gcd(F_n+4,F_{n+1}+4) \qquad\text{for every }n\ge16.

More generally, there is a complete local criterion for an arbitrary integer shift:

\gcd(K_n+a,K_{n+1}+a)\le2 \quad\text{for every }n\ge2

if and only if

a\not\equiv2\pmod4 \quad\text{and}\quad a\not\equiv-K_p\pmod p \quad\text{for every odd prime }p.

Indeed,

\gcd(K_n+a,K_{n+1}+a)=\gcd(K_n+a,n!).

For any odd prime $p$ and $n\ge p$ , factorial stabilization gives $K_n\equiv K_p\pmod p$ ; hence $p$ divides this gcd exactly when $a\equiv-K_p\pmod p$ . Likewise $K_n\equiv2\pmod4$ for $n\ge4$ , so divisibility of the gcd by $4$ occurs exactly when $a\equiv2\pmod4$ . These exhaust all ways the gcd can exceed $2$ .

The left-factorial values and the residue $K_{17}\bmod17=13$ are pre-existing data in OEIS A003422 and OEIS A100612, respectively. Applying that residue refutes both Conjecture 2 and the preceding shift- $4$ assertion in arXiv:2509.06077. The separate unshifted case $a=0$ remains the open Kurepa conjecture and is not claimed here.