Erdős Problem #276

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 terms are positive integers satisfying the stated second-order recurrence. The extra requirement is that no fixed $M\geq2$ has $\gcd(M,a_j)>1$ for every $j$. Equivalently, no finite set of primes divides between them every term; it is stronger than requiring the gcd of all terms to be one. [S1] [S2]

Statement recorded in the supplied source.

Is there an infinite Lucas sequence $a_0,a_1,\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Prove for the Ismailescu-Son sequence or another all-composite Lucas sequence that every finite prime set misses at least one term. [S1]

Short English statement

Can a Fibonacci-type recurrence have only composite terms without any fixed integer sharing a nontrivial factor with every term?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2025 (EP-276), 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 276, https://www.erdosproblems.com/276. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Dan Ismailescu and Jaesung Son, A New Kind of Fibonacci-Like Sequence of Composite Numbers, Journal of Integer Sequences 17 (2014), Article 14.8.2. https://www.maths.tcd.ie/EMIS/journals/JIS/VOL17/Ismailescu/ism8.pdf. Bibliographic information from the supplied record.

• S4 Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 27. https://combinatorica.hu/~p_erdos/1979-07.pdf. Bibliographic information from the supplied record.

• S5 Graham, R. L., A Fibonacci-Like Sequence of Composite Numbers. Math. Mag. (1964), 322-324. Bibliographic information from the supplied record.

• S6 Ismailescu, Dan and Son, Jaesung, A new kind of {F}ibonacci-like sequence of composite numbers. J. Integer Seq. (2014), Article 14.8.2, 9. Bibliographic information from the supplied record.