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

The question

conjecture1. What is the smallest integer $m>1$ such that $a(10^m)$ is nonzero?

• _Farideh Firoozbakht_, Jan 07 2015

conjecture2. Conjecture: If $n$ is not of the form $10^m$ then $a(n)$ is nonzero.

• _Farideh Firoozbakht_, Jan 07 2015

conjecture3. What is the smallest odd prime $p$ such that $(10^{p^2}-1)/(10^p-1)$ is a prime number (and $a(10^{p-1})$ could be nonzero)?

• _Farideh Firoozbakht_, Jan 07 2015

$a(n)$ 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 $n$.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«86766» (3 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem conjecture1 :
    answer(sorry) =
      if h : ∃ m, 1 < m ∧ a (10 ^ m) ≠ 0 then
        some (sInf {m | 1 < m ∧ a (10 ^ m) ≠ 0})
      else
        none

theorem conjecture2 (n : ℕ) (hn : 0 < n) (h : ∀ m : ℕ, n ≠ 10 ^ m) : a n ≠ 0

theorem conjecture3 :
    answer(sorry) =
      if h : ∃ p, p.Prime ∧ 2 < p ∧ ((10 ^ (p ^ 2) - 1) / (10 ^ p - 1)).Prime then
        some (sInf {p | p.Prime ∧ 2 < p ∧ ((10 ^ (p ^ 2) - 1) / (10 ^ p - 1)).Prime})
      else
        none

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• A086766

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.