Erdős Problem #349

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.

In contrast with the preceding multiset entry, the source specifies distinct integer values. Take the set of values $\{\lfloor t\alpha^j\rfloor:j\geq0\}$, ignore zero when forming positive sums, and use each value at most once. The index convention is recorded explicitly; adding or removing initial values can matter for completeness. [S1] [S2]

Statement recorded in the supplied source.

For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\lfloor t\alpha^n\rfloor$)? [S1]

Scope and interpretation.

The source does not print the initial exponent. The convention used here includes exponent zero; the exact indexing should be retained when comparing delicate deletion variants. [S1]

Short English statement

For which positive scales and bases do distinct values of a floored geometric sequence represent every sufficiently large integer by finite sums?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2067 (EP-349), 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 349, https://www.erdosproblems.com/349. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Graham, R. L., On a conjecture of Erdős in additive number theory. Acta Arith. (1964/65), 63-70. Bibliographic information from the supplied record.