Factorial Divisibility and Logarithmic Gaps

• Factorial divisibility with logarithmic gaps is a phenomenon where divisibility relations in factorials or products of consecutive integers are separated by gaps that grow logarithmically.
• Analytic tools including linear forms in logarithms, sieve methods, and p-adic analysis establish these minimal gap results, highlighting the sparseness of such configurations.
• These findings have practical implications for combinatorial, Diophantine, and additive problems, providing explicit bounds and constants derived from prime number theory and approximation techniques.

Factorial divisibility with a logarithmic gap refers to the phenomenon that, in a variety of central combinatorial, Diophantine, and additive problems involving factorials, products of consecutive integers, or related forms, the combinatorial divisibility or representability properties cannot occur too densely—they are separated by gaps whose size is governed by logarithmic or iterated-logarithmic functions. This principle underlies results quantifying the minimum possible gap or difference between objects such as consecutive products, factorizations of factorials, or the difference between factorials and certain exponential forms. The optimality and precise formulation of such gaps rely on a detailed synthesis of analytic number theory—most notably, effective linear forms in logarithms, sieve methods, and $p$-adic valuation arguments.

1. Fundamental Definitions and Canonical Problems

The term originates in several research lines, notably those concerned with whether one combinatorial or arithmetical object divides another, or whether a given structure can be realized subject to size constraints. Three canonical problems exemplify this theme:

• Divisibility of products of consecutive integers: When does $a(a+1)(a+2)\mid b(b+1)(b+2)$? A fundamental result is that such divisibility cannot occur for $b$ only slightly larger than $a$; indeed, $b$ must be substantially larger, with a lower bound governed by a power of $a$ and logarithmic terms (Chan, 2024).
• Decomposition of factorials into large factors: Given $N!\;$, what is the largest possible $t(N)$ such that $N!$ can be written as the product of $N$ integers, each at least $t(N)$? The answer reveals a gap of size $N/\log N$ below the 'trivial' threshold $N/e$ (Alexeev et al., 26 Mar 2025).
• Divisibility with a logarithmic window: For the divisibility $a! b! \mid n! (a+b-n)!$, infinitely many triples $(a,b,n)$ exist with the difference $a+b-n$ lying in a prescribed logarithmic window, i.e., $C_1 \log n < a+b-n < C_2 \log n$ for arbitrary positive $C_1 < C_2$ (Sothanaphan, 12 Jan 2026).

These problems encode, in distinct but related ways, the impossibility of dense or tight configurations and reveal the essential role of logarithmic growth laws.

2. Theorems Quantifying Logarithmic Gaps

A sequence of sharp theorems has established the presence of explicit logarithmic or iterated-logarithmic divisibility gaps:

• Gap theorem for triple products: If $a(a+1)(a+2)\mid b(b+1)(b+2)$ for $a<b$, then

$b \gg (a\log a)^{1/6}(\log\log a)^{1/3}$

for an absolute constant, indicating a strong non-density of such divisibility relations (Chan, 2024).

• Factorial decomposition threshold: The maximal $t(N)$ such that $N! = a_1\cdots a_N$ with $a_i\geq t(N)$ satisfies

$$\frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$

with $c_0 \approx 0.30441901$ explicit, manifesting a logarithmic gap below the naive $N/e$ regime (Alexeev et al., 26 Mar 2025).

• Logarithmic window for factorial divisibility: For any $0<C_1<C_2$, there exist infinitely many triples $(a,b,n)$ with $a!b!\mid n!(a+b-n)!$ and

$C_1\log n < a+b-n < C_2\log n,$

showing that the difference cannot be arbitrarily small nor must it diverge faster than logarithmically (Sothanaphan, 12 Jan 2026).

• Distance to repunits: If $n!-(b^p-1)/(b-1) \ge 0$, then for large $n$,

$$n! - \frac{b^p - 1}{b-1} \ge \frac{1}{2}\frac{\log\log n}{\log\log\log n},$$

providing a doubly-logarithmic lower bound on the proximity between factorials and repunits (Filaseta et al., 2024).

These theorems are unconditional, except in certain cases where conditional improvements (such as under the abc conjecture) yield stronger, power-type bounds.

3. Methods: Sieve, Diophantine Approximation, and $p$-adic Analysis

The proofs employ a mélange of advanced techniques adapted to the combinatorial and analytic structure of the underlying problem:

• Linear forms in logarithms: The gap theorems for products of consecutive integers are derived by recasting divisibility conditions as questions about rational approximations to algebraic numbers (e.g., $t^{1/3}$), then invoking effective Liouville-Baker-Feldman bounds to show that such approximations are only possible with a logarithmic gap in the denominators (Chan, 2024).
• Refined combinatorial and sieve methods: The problem of decomposing $N!$ into large factors is resolved via a detailed analysis of the distribution of prime factors, matching lower and upper bounds through the use of the prime number theorem, combinatorial swaps, and greedy bipartite matching for large primes; the error terms are controlled via density estimates for primes and combinatorial arrangements (Alexeev et al., 26 Mar 2025).
• $p$-adic valuation and Kummer’s theorem: The factorial divisibility within a logarithmic window is addressed by reducing the problem to the divisibility of binomial coefficients and analyzing $p$-adic valuations via carry counts in base-$p$ expansions. A probabilistic threshold argument identifies "carry-rich but spike-free" instances, leveraging Chernoff bounds and residue class estimates to guarantee existence in a given window (Sothanaphan, 12 Jan 2026).

4. Explicit Constants, Lower-Order Terms, and Computability

Precise quantitative results in this area depend critically on explicit constants and control of error terms:

• Explicit constants: For factorial decompositions, $c_0$ is given by

$$c_0 = \frac{1}{e}\int_0^1 \left\lfloor \frac{1}{x} \right\rfloor \log \left( e x \left\lceil \frac{1}{e x} \right\rceil \right) dx \approx 0.30441901$$

arising from optimization of the logarithmic contribution of prime intervals near $N/e$ (Alexeev et al., 26 Mar 2025).

• Lower-order error: The $O(1/\log^{1+c}N)$ error in $t(N)$ can be reduced by sharper prime number theorem estimates, careful combinatorial matching, and refined saddle-point estimates over dyadic ranges of primes.
• Numerical thresholds: The Guy–Selfridge $1/3$-barrier is confirmed for $N\geq 43632$, and stronger explicit bounds are obtainable via computer-assisted methods (Alexeev et al., 26 Mar 2025).
• Ineffective constants: Certain thresholds (e.g., the $n_0$ in the theorems involving distance to repunits) are not given explicitly, as their determination rests on ineffective components of the prime ideal theorem or zero-free regions of L-functions (Filaseta et al., 2024).

5. Conditional Results and Open Problems

The scope and sharpness of logarithmic-gap results are subject to conditional improvements and unresolved questions:

• Conditional enhancements: Under the abc conjecture or related Diophantine assumptions, logarithmic gaps may be replaced by near-linear or power-type bounds. For example, for $a^2(a^2+\ell)\mid b^2(b^2+\ell)$, the bound improves (conditionally) from $(a\log a)^{1/12}(\log\log a)^{1/2}$ to $a^{8/7-\epsilon}$ (Chan, 2024).
• Sharpness of exponents: The exponents (e.g., $1/6$ for consecutive products) are dictated by the quantitative strength of Baker–Feldman estimates and are not expected to be optimal. It is widely believed that genuinely linear gaps (i.e., $b\gg ca$ for some $c>1$) hold in some cases, but no known method attains them unconditionally.
• Higher-degree and general polynomial extensions: For general polynomials, only partial classifications of gap orders are known. A complete and quantitatively sharp description remains open except in low-degree settings (Chan, 2024).

A plausible implication is that further advances in Diophantine approximation, sieve theory, or arithmetic geometry could substantially tighten these bounds.

6. Connections, Historical Context, and Related Phenomena

These results integrate and extend a long sequence of investigations into the arithmetic of factorials, combinatorial divisibility, and representation by products. Techniques draw from classic works of Erdős, Graham, Selfridge, Choi-Lam-Tarnu, and the analytic machinery of Lagarias–Montgomery–Odlyzko for least-prime estimates (Filaseta et al., 2024). Many problems have their origin in longstanding conjectures and have been motivated by questions on the structure of binomial coefficients, decomposability of integers, and the distribution of primes.

The proven logarithmic-gap phenomenon provides a unifying perspective for a variety of previously disconnected problems, demonstrating the rigidity and sparseness of dense divisibility relations in factorial and polynomial products.

7. Summary Table: Principal Results

Problem Class	Best Proven Gap	Reference
$a(a+1)(a+2)\mid b(b+1)(b+2)$	$b\gg (a\log a)^{1/6}(\log\log a)^{1/3}$	(Chan, 2024)
Max $t(N)$ with $N!$ as $a_1\dots a_N$	$t(N)/N = 1/e - c_0/\log N + O(\cdots)$	(Alexeev et al., 26 Mar 2025)
$a!b!\mid n!(a+b-n)!$	$C_1\log n< a+b-n < C_2\log n$	(Sothanaphan, 12 Jan 2026)
$n! - R_p(b)\geq 0$	$$n! - R_p(b) \geq 0.5\frac{\log\log n}{\log\log\log n}$$	(Filaseta et al., 2024)

These results collectively establish that strong arithmetic divisibility or representation phenomena in factorial-related settings are separated by quantifiable logarithmic or iterated-logarithmic minimal gaps, constrained by the analytic structure of the problem and, in some cases, by currently inaccessible Diophantine principles.