The board / number theory / #690
Erdős Problem #690
SOLVEDdatabase updated 2025-08-31
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The set of positive integers whose `k`-th smallest prime factor is `p`: if \(p_1<p_2<\cdots\) are the primes dividing \(n\) then \(p_k=p\).
A function on the primes is *unimodular* if it first increases in \(p\) until its maximum then decreases.
Let \(d_k(p)\) be the density of those integers whose \(k\)th smallest prime factor is \(p\) (i.e. if \(p_1<p_2<\cdots\) are the primes dividing \(n\) then \(p_k=p\)). For fixed \(k\geq 1\) is \(d_k(p)\) unimodular in \(p\)? That is, it first increases in \(p\) until its maximum then decreases. The answer is no in general: Cambie [Ca25] has shown that \(d_k(p)\) is unimodular for \(1\leq k\leq 3\) and is not unimodular for \(4\leq k\leq 20\). The densities \(d_k(p)\) exist (see `erdos_690.variants.hasDensity`), so the statement quantifies over any function `d` recording them.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/690.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/690
Attempts
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