Multiplicative order of 2 mod $2n+1$

formal-conjectures · oeis · ams-11

The multiplicative order of 2 modulo $2n+1$.
In other words, the least $m > 0$ such that $2n+1$ divides $2^m - 1$.

If $p$ is an odd prime then $a((p^3-1)/2) = p \cdot a((p^2-1)/2)$.
Because otherwise $a((p^3-1)/2) < p \cdot a((p^2-1)/2)$ iff $a((p^3-1)/2) = a((p-1)/2)$
for a prime $p$. Equivalently $p^3$ divides $2^{p-1}-1$, but no such prime $p$ is known.
- Thomas Ordowski, Feb 10 2014

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/A002326
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/2326.lean
Local target: Corpus.OEIS2326.conjecture1
Source SHA-256: c0ac17e5813dc4d83b227d6afcef08f332b0634126bc35dc809fa0c6b61a2859
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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