Erdős Problem #820

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

The bases are integers $2\leq k<l$. This is the nontrivial base convention corroborated by the source's explicit equivalence involving $2^n-1$ and $3^n-1$. All gcds are positive greatest common divisors. [S1] [S2]

Statement recorded in the supplied source.

Let $H(n)$ be the smallest integer $l$ such that there exist $k<l$ with $(k^n-1,l^n-1)=1$. Is it true that $H(n)=3$ infinitely often? (That is, $(2^n-1,3^n-1)=1$ infinitely often?) Estimate $H(n)$. Is it true that there exists some constant $c>0$ such that, for all $\epsilon>0$, $ H(n) > \exp(n^{(c-\epsilon)/\log\log n}) $ for infinitely many $n$ and $ H(n) < \exp(n^{(c+\epsilon)/\log\log n}) $ for all large enough $n$? Does a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Independently verify and publish the lower-bound deduction; settle H(n)=3 infinitely often, prove a matching all-large-n upper bound and constant, and handle the fixed-base variant. [S1]

Short English statement

How large must the smaller pair of bases be to make their nth powers minus one coprime, and how often do bases two and three suffice?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2321 (EP-820), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 820, https://www.erdosproblems.com/820. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Leonard M. Adleman, Carl Pomerance, and Robert S. Rumely, On distinguishing prime numbers from composite numbers, Annals of Mathematics 117 (1983), 173-206. https://annals.math.princeton.edu/1983/117-1/p07. Bibliographic information from the supplied record.

• S4 Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202. Bibliographic information from the supplied record.