Smallest $r$ such that (concatenation of $n$, $r$ times) $\cdot 10 + 1$ is prime

formal-conjectures · oeis · ams-11

$a(n)$ is the smallest $r$ where (concatenation of $n$, $r$ times with itself) $\cdot 10 + 1$
is a prime, or $0$ if no such number exists.
The number resulting from concatenating $n$, $r$ times, is
$n \cdot \sum_{i=0}^{r-1} (10^d)^i$, where $d$ is the number of digits of $n$.

Conjecture: If $n$ is not of the form $10^m$ then $a(n)$ is nonzero.
- _Farideh Firoozbakht_, Jan 07 2015

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A086766
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/86766.lean
Local target: Corpus.OEIS86766.conjecture2
Source SHA-256: ac371778619224516ecbd7c6ad030cc819db20f4c8e9b367e12d93038bc06373
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

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