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Erdős Problem #276

OPENdatabase updated 2025-08-31

No proof or disproof is known, and it isn't known whether a finite computation could settle it.

What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).

We define a Lucas sequence to be a Fibonacci sequence with arbitrary starting points `L 0` and `L 1`. TODO: There seems to be multiple definitions in the literature, some of which also allow coefficients in the recurrence relation. For now this simple definition has been chosen as it agrees best with the Erdős problem in this same file. However before moving this into `ForMathlib` one should make a conscious decision about which definition to choose.

Is there an infinite Lucas sequence \(a_0, a_1, \ldots\) where \(a_{n+2} = a_{n+1} + a_n\) for \(n \ge 0\) such that all \(a_k\) are composite, and yet no integer has a common factor with every term of the sequence?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/276.
Prizenone
Topicsnumber theory, covering systems
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/276

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the Ismailescu--Son candidate is re-certified, and any covering integer must contain at least 59 distinct primes above 10^7 (and outside its 30 covering primes), but ruling out every finite cover rema

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