Let A(n)=a:1≤ a 1, |A(n)|≤ 3. The bound is equivalent to the assertion that any two chains mathfrakC_k_1 and mathfrakC_k_2, with k_1>k_2≥ 3, do not share an integer. The source presents this as an open question motivated by the observed size property.

The size bound for the set of mutual divisibility solutions

Let

\mathcal{A}(n)=\{a:1\leq a<n,\ n\mid a^2-1,\ a\mid n^2-1\}.

Size-bound conjecture. For any integer $n>1$ ,

|\mathcal{A}(n)|\leq 3.

The bound is equivalent to the assertion that any two chains $\mathfrak{C}_{k_1}$ and $\mathfrak{C}_{k_2}$ , with $k_1>k_2\geq 3$ , do not share an integer. The source presents this as an open question motivated by the observed size property.

References

Primary source

Srikanth Cherukupally, “On the size of \a: 1a<n, n|a^2-1, a|n^2-1\ for number n”, arXiv:2603.17434 (2026).

divisibility number-theory

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Open

The conjecture remains open: no proof or counterexample was found showing whether this set can ever have more than three elements.

The conjecture asks whether the set of integers satisfying both divisibility conditions always has at most three elements for every integer $n>1$ . The source explicitly presents this as an open question based on observed examples.

Current status (as of September 2026): The bound $|\mathcal{A}(n)|\leq 3$ remains unsettled, with no publicly recorded proof, counterexample, or substantive progress found.

Sources

No solutions have been posted yet.