slug: oeis-a390088-prime-digit-base-existence bibkey: oeis2025a390088 doi: null url: https://oeis.org/A390088 triage: theorem motivation_gids:

• D5/S1/Digit/PrimeDigitBaseClassification

The A390088 exception set

Problem

For each number, ask whether some base above one writes it with all digits prime. Felix Huber conjectured on October 29, 2025 that exactly five numbers admit no such base.

Motivation

This is a first-tier recent OEIS conjecture. Direct inspection on September 9, 2026 found it still labelled Conjecture, with the exception set stated verbatim and the sequence marked easy.

Gap

The entry states the exception set but supplies no argument. The searched indexes returned no proof.

Route

Above ten, two certificates suffice. For an even argument take the base that is half of two less than it, giving the two-digit representation with both digits two; for an odd argument take the base that is half of three less than it, giving digits three and two. Both digits are prime and both are below their base, so the certificate is valid in each case.

Below ten the question is finite and is settled by inspection. Four arguments admit no base; one is excluded because its digit list is empty in every base, which would otherwise make a universally quantified condition on digits vacuously true; and four admit a base, one of them only a single base.

For the nonexistence direction, split on whether the argument is below the base. If it is, the digit list is the argument itself and primality is direct. If it is not, the base is below ten as well and a bounded enumeration over pairs settles it.

Falsifier

A number outside the five with no base writing it in prime digits, or one of the five with such a base, would contradict the theorem about the defined condition.

Evidence

• Module: D5/S1/Digit/PrimeDigitBaseClassification.lean.
• Main theorem: a390088, an equivalence for every argument.
• Supporting result: prime_digit_base_certificate, the two certificates.
• The finite reasoning is confined to bounded enumerations over pairs below ten and to the five explicit small arguments; the half of the statement ranging over all arguments is a general argument with no enumeration.
• There is no finite cutoff in the public statement.

The caller verified before dispatching an implementation seat. The exception set computed directly from the definition, searching bases to four hundred for arguments below four hundred, was exactly the five. Every argument from ten to three thousand was covered by one of the two certificates, with no exception. The four small arguments that admit no base were checked against every base below their own value, and the one small argument admitting only a single base was identified.

Triage

theorem. The equivalence is proved for every argument. The least such base, which is what the sequence records, is not formalized; only the existence question the conjecture asks about.

ASSUMED-UNVERIFIED

First-publication priority is not established. The searches do not exclude private or unindexed proofs. The identification with the OEIS entry is documentary; the kernel verifies the explicitly defined condition.