bibkey: oeis2025a390088 authors: OEIS Foundation Inc. year: 2025 title: OEIS A390088 doi: null url: https://oeis.org/A390088 claim: A comment on OEIS A390088 conjectures that exactly five numbers admit no base in which all their digits are prime. strata_touched:
- D5/S1/Digit/PrimeDigitBaseClassification license: citation-only triage: anchor
OEIS A390088
The entry assigns to each number the least base above one in which all of its digits are prime, or zero when no such base exists. Felix Huber added a comment on October 29, 2025 conjecturing that the zero value occurs at exactly five arguments. The caller found it still labelled Conjecture on September 9, 2026, with the most recent revision touching only status and programs.
Why the classification is short
Every argument of at least ten admits a base, and only two certificates are needed. An even argument is two more than twice the base when the base is half of two less than it, giving digits two and two; an odd argument is three more than twice the base at the corresponding base, giving digits three and two. Both two and three are prime, and both bases exceed the digits they carry.
Below ten the question is finite. Four of the arguments admit no base, one is excluded because the digit list is empty, and the remaining four admit one. One of those four admits only a single base, which is the case most easily missed when the argument is split into a general half and a finite half.
What the Lean module proves
D5/S1/Digit/PrimeDigitBaseClassification states the equivalence for every
argument: a base with all digits prime exists exactly when the argument is
none of the five. The nonexistence direction splits on whether the argument is
below the base, where the digit list is a singleton and primality is direct,
or not, where a bounded enumeration over pairs below ten settles it. The
existence direction uses the two certificates above ten and explicit bases
below.
The finite reasoning is confined to bounded enumerations; the half of the statement that ranges over all arguments is a general argument.
ASSUMED-UNVERIFIED: the identification of the printed sequence with the module’s predicate is a human reading of the entry, not a machine proof that the two denote the same condition. The module does not formalize the least such base, only the existence question the conjecture asks about.
Search log
- Caller reading, 2026-09-09: the seat reported checking the entry and its revision history, finding the conjecture and the exception set stated verbatim and the sequence marked easy, with no later revision adding a proof.
- The seat’s receipt is
所查来源未找到, explicitly not a claim that no proof exists anywhere. - Caller verification, 2026-09-09: the orchestrator computed the exception set directly and checked the two certificates independently before dispatching a seat. Readings are in the problem entry. The seat also flagged that the empty digit list of zero makes a universally quantified digit condition vacuously true, so the argument must be excluded explicitly; the orchestrator confirmed this against the library’s convention.
Verified locator
- URL: https://oeis.org/A390088