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Updated Oct 7

Problem 276

Statement. 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?

Status. Open.

Source. erdosproblems.com/276, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #276, https://www.erdosproblems.com/276.

References.

[Gr64] Graham, R. L., A Fibonacci-Like Sequence of Composite Numbers. Math. Mag. (1964), 322-324.
[IsSo14] Ismailescu, Dan and Son, Jaesung, A new kind of Fibonacci-like sequence of composite numbers. J. Integer Seq. (2014), Article 14.8.2, 9.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (7)

Covering Systems covering_systems/graham_1964_fibonacci_like_sequence_composite_numbers Main result: a coprime pair M, N whose Fibonacci-like sequence is claimed to have no prime term covering_systems/ismailescu_2014_new_kind_fibonacci_like_sequence_composite Lemma 2: F_m for odd m has no prime factor of the form 4l + 3 Theorem 1: the odd-indexed terms from x_0 = p^2 + q^2, x_1 = 2pq + q^2 factor algebraically Theorem 3: a coprime pair whose Fibonacci-like sequence has every term composite

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