The board / number theory / #672
Erdős Problem #672
VERIFIABLEdatabase updated 2025-08-31
Still open — but if it is true, a finite example proves it, so a computer search could in principle prove it. A search can't disprove it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Erdős problem 672 conjectures that the below holds for any \(k ≥ 4\) and \(l > 1\).
Can the product of an arithmetic progression of positive integers \(n, n + d, ..., n + (k - 1)d\) of length \(k ≥ 4\), with \((n, d) = 1\), be a perfect power? Erdős believed not, i.e. that `Erdos672With k l` holds for all \(k ≥ 4\) and \(l > 1\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/672.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/672
Attempts
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Modulo verified square/cube/fifth-power theorems, any counterexample at the first unresolved lengths k=35..38 has prime exponent at least 7 and largest term at least 13,357,343; 13,750,206; 47,045,881
PARTIAL