Separable Powers of Two in Divisor Theory

• Separable powers of two are integers of the form 2^k that participate in interlocking divisor pairs, exhibiting a unique combinatorial structure.
• Density analyses reveal that while many 2^k are separable, there exists a positive lower density of exponents for which they are not, for example when k ≡ 1, 2, 9, or 10 (mod 12).
• Explicit constructions and interlocking conditions involving alternating divisor sequences provide profound insights with implications for additive number theory and combinatorial divisor spacing.

A separable power of two is a positive integer of the form $2^k$ that satisfies a combinatorial property defined via interlocking pairs of positive integers, in the sense introduced by Erdős and Hall. In this framework, two integers $(m,n)$ are interlocking if the ordered lists of their divisors (ignoring 1) alternate: between every pair of consecutive divisors (greater than 1) of one, there is a divisor of the other. An integer is said to be separable if it appears in such an interlocking pair. This concept interfaces with broader studies of additive number theory, the combinatorial and arithmetic structure of powers of two, and density results concerning divisibility patterns.

1. Formal Definition of Separable Powers of Two

Let $n=2^k$ and $m$ be positive integers. Denote the ordered divisors of $n$ (excluding 1) as $d_1 < \cdots < d_{\tau(n)}$, and similarly for $m$ with $d_1' < \cdots < d_{\tau(m)}$. A pair $(m,n)$ is interlocking if, between each consecutive pair $d_{j-1}', d_j'$ of $n$'s divisors (with $d_{j-1}', d_j' > 1$), there exists a divisor $d_\ell$ of $m$ satisfying $d_{j-1}' < d_\ell < d_j'$, and likewise for $(n,m)$. A number is separable if it belongs to such a pair.

Powers of two are examined in this context by considering $n = 2^k$ and studying, for varying $k$, whether such an $m$ exists for which $(m, 2^k)$ interlocks. The density and structural properties of those $k$ for which $2^k$ is separable, as well as those for which it is not, are central questions.

2. Density Results and Resolution of Erdős–Hall Conjectures

The lower density of separable powers of two (that is, the asymptotic proportion of exponents $k$ for which $2^k$ is separable) is shown to be positive, with an explicit construction establishing that for $k$ in a suitable subset, one can always find $m$ interlocking with $n=2^k$. Conversely, the paper (Cambie et al., 22 Oct 2025) demonstrates that there is also positive lower density of exponents $k$ for which $2^k$ is not separable. Specifically, for $k \equiv 1,2,9,10 \pmod{12}$ and $k > 2$, $2^k$ is not separable.

This refutes the conjecture of Erdős and Hall that "almost all" powers of two are separable, while positive lower density for separable exponents confirms the existence of infinitely many such powers. The methods involve detailed analysis of the divisor functions and the interaction of divisor spacings between powers of two and auxiliary constructed numbers, often leveraging explicit bounds from prime gap theory and combinatorial decomposition.

3. Interlocking Pair Structure: Divisor Alternation and Constraints

For interlocking pairs $(m,n)$, the key technical constraints are governed by the value of the divisor-counting function $\tau(n)$.

• If $n < m$, then $\tau(m) = \tau(n)$.
• If $n > m$, then $\tau(m) = \tau(n) - 1$.

Analysis of these cases, together with congruence information determined by the construction (e.g., considerations based on residue classes of $k$ modulo 12), establishes when the interlocking is impossible and when explicit constructions ensure success. The construction of $m$ typically involves products of primes chosen to interleave with the divisor set of $2^k$, under spacing conditions derived from explicit bounds in (Cambie et al., 22 Oct 2025).

In the positive density result for separable exponents, $k$ is written as $2^t s$ for $t$ a fixed integer and $s$ in a set $S$ defined by divisor distributions and prime selection, with $m=231\prod_{i=4}^r p_i^{e_i}$ pairing accordingly.

4. Finiteness of Interlocking Pairs with Special Products

A further conjecture by Erdős and Hall, concerning the finiteness of interlocking pairs $(m,n)$ such that $mn$ is equal to the primorial $P_k$ (product of the first $k$ primes), is resolved affirmatively in (Cambie et al., 22 Oct 2025). Specifically, for large $k$, no such interlocking pair exists; the proof relies on incompatibility of forced divisor structure and exhaustive combinatorial conditions as the number of primes in the product grows.

5. Implications for Combinatorial and Additive Number Theory

These results connect the structure of separable powers of two with the fine-grained analysis of divisor functions and combinatorial alternation, demonstrating that even for highly structured sets like powers of two—whose divisors are exactly the set $\{2^j: 0 \leq j \leq k\}$—the possibility of interlocking hinges on nontrivial combinatorial and arithmetic constraints. The positive density phenomena indicate a balance: separability is neither ubiquitous nor rare in this family.

The density results have broader implications, suggesting that the space of numbers admitting interlocking pairs is rich but strongly modulated by arithmetic properties. The techniques, including explicit construction and analytical density estimates, offer methodologies that may be transferable to related combinatorial divisor-interleaving or alternation problems.

6. Key Formulas and Technical Results

Notation/Formula	Description	Context
$1 = d_1 < \cdots < d_{\tau(m)} = m$	Ordered divisors of $m$	Used in definition of interlocking
Between $d_{j-1}', d_j'$ (of $n$), exists $d_\ell$ of $m$ with $d_{j-1}' < d_\ell < d_j'$	Divisor alternation condition	Interlocking definition
$\tau(m) = \tau(n)$ or $\tau(m) = \tau(n)-1$	Relationship of numbers of divisors	Lemma in (Cambie et al., 22 Oct 2025)
$k \equiv 1,2,9,10 \pmod{12},\ k>2$	Non-separable exponents	Theorem 2.1 (Cambie et al., 22 Oct 2025)
$m=231\prod_{i=4}^r p_i^{e_i}$	Explicit construction for $m$ in interlocking pair	Positive density of separable $k$

In sum, the concept of separable powers of two elucidates the interplay between prime factorization, combinatorial divisor alignments, and structural properties of integer sequences. The resolution of the conjectures of Erdős and Hall via explicit density computations and careful arithmetic analysis enriches the theory of separable numbers and interlocking pairs.