The board / number theory / #1180
Erdős Problem #1180
PROVEDdatabase updated 2026-03-06
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`Represents ε p a s` says that the multiset `s` of denominators `n` with `1 ≤ n ≤ p^ε` and `n` coprime to `p` has `∑ n⁻¹ = a` in `ZMod p`.
`C ε` is the least `C` such that, for all primes `p`, every residue modulo `p` is the sum of at most `C` elements of `{n⁻¹ : 1 ≤ n ≤ p^ε}` (or `0` if no such `C` exists).
Let \(\epsilon>0\). Does there exist a constant \(C_\epsilon\) such that, for all primes \(p\), every residue modulo \(p\) is the sum of at most \(C_\epsilon\) many elements of \[\{ n^{-1} : 1\leq n\leq p^\epsilon\}\] where \(n^{-1}\) denotes the inverse of \(n\) modulo \(p\)? The original question was answered in the affirmative by Shparlinski [Sh02], who proved that \(\ll \epsilon^{-3}\) summands suffice for all sufficiently large primes. This was improved to \(\ll \epsilon^{-2}\) by Glibichuk [Gl06]. It is trivial that at least \(\gg \epsilon^{-1}\) summands are required, and it may be that \(C_\epsilon\leq \epsilon^{-1-o(1)}\) is possible. See also [540](https://www.erdosproblems.com/540).
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/1180.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1180
Attempts
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