trureturing — Discovering truth

bibkey: bala2026a126156
authors: Peter Bala
year: 2026
title: "OEIS A126156: eventual periods dividing the Euler totient"
doi: null
url: https://oeis.org/A126156
claim: "Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic with the period dividing phi(k), where phi(n) = A000010(n). An example is given below. - _Peter Bala_, Mar 01 2026"
strata_touched:
  - D5/S1/Recurrence/Periodic/SquareRootSecantTotientPeriodRefutation
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A126156

OEIS A126156 revision 110, NAME and third COMMENTS entry; there are no page numbers, DOI, source arXiv identifier or numbered paper conjecture. The author field credits Bala for the conjecture rather than the entire entry. The source entry and its internal form are identified in issue #22077.

Source statements

Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.

Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic with the period dividing phi(k), where phi(n) = A000010(n). An example is given below. - Peter Bala, Mar 01 2026

E.g.f.: Sum_{n>=0} a(n)x^(2n)/(2*n)! = sqrt(sec(sqrt(2)*x)). - David Callan, Jan 03 2011

The offset is zero. The rational series A has constant coefficient one and satisfies A² cos(√2 x) = 1, with the cosine coefficient at an even j equal to (-2)^(j/2)/j! and at an odd j equal to zero. The sequence term is (2n)! [x^(2n)] A; its factorial expression is used directly in the universal period assertion. The triangle A126155 is an alternate source representation; the formal argument uses the EGF and does not assume a triangle identity.

Mathematical boundary

The repository derivation excludes every eventual onset for a positive period dividing φ(4)=2. Integer derivative polynomials give two-step differences congruent to two modulo four; parity follows by induction. This refutes the universal period-divides-totient assertion. A universal bound of 2φ(k) and a general odd-modulus theorem remain open here.

The literature screen in issue #22077 found no settlement in its inspected sources. Completeness is ASSUMED-UNVERIFIED: rate-limited searches and unread linked full texts leave prior-art coverage incomplete. No worldwide priority claim is made, and no result in the inspected moment papers was identified as depending on this conjecture.

The independent finite checks are in the private repository the-omega-institute/trureturing-experiments, commit c2e136ef37ea6d8610d3bdb475f70a20fc3515e3, file docs/reports/bala-a126156-totient-period/check.py, SHA-256 1d2ea054dbc8cb7285102d90275b218c13553923476f1393efa546e0e33e1650. The recorded command python3 docs/reports/bala-a126156-totient-period/check.py exited 0 with ALL_CHECKS_OK. It checks the residue cycle through n=200 and the displayed modulus-55 example with shift 20 on n=1..180; these are finite computations, separate from the all-index formal argument.