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Preprint · Oct 2026

Searching for a Decreasing Function in the Collatz Conjecture

This paper investigates whether a Lyapunov-style function can be found that decreases along Collatz trajectories, and concludes that no such function is established here: the work is a structured research programme, not a proof. Starting from the failed candidate F(n) = n, the logarithmic potential F(n) = log₂ n reveals an average downward drift once the process is compressed to odd integers; residue classes modulo powers of 2 then explain how many divisions by 2 follow each odd step, and a residue-dependent correction on classes modulo a power of 2 turns part of the problem into an optimisation problem. The aim is to show how an elementary-looking problem becomes a serious investigation spanning dynamical systems, probability, number theory, optimisation and computation.

Taha Tanver Athar, Tanver Athar18 reads
Preprint · Sep 2026

When an Infinite Hotel Runs Out of Rooms

The infinite hotel is a useful setting for examining the counterintuitive properties of infinite sets. A hotel with countably infinitely many rooms can accommodate additional guests even when every room is occupied, because a countably infinite set can be placed in bijection with certain proper subsets of itself. This phenomenon does not, however, imply unlimited cardinal capacity. We formally distinguish accommodation from room occupancy, and show that a countably infinite hotel has capacity ℵ₀ and can accommodate any guest set of cardinality at most ℵ₀ under the one-guest-per-room rule. An uncountable guest set, such as one indexed by the real numbers, has the cardinality of the continuum, which Cantor's theorem shows to be strictly greater than ℵ₀ — hence no injection from the guests into the rooms exists, and complete accommodation is impossible. An infinite hotel can run out of rooms not because it has a final room, but because the arriving population has strictly greater cardinality than the room set.

Taha Tanver Athar, Tanver Athar23 reads
Preprint · Jul 2026

The Echo-Square Coprime sequence: a new integer sequence with index-dependent arithmetic and digit-sum constraints

We introduce the Echo-Square Coprime (ESC) sequence, a greedy smallest-unused-integer construction binding each term to its own index through three simultaneous constraints: coprimality with n, a triangular digit sum, and a square echo requiring a(n) to be a perfect square exactly when n is. The conjunction produces sharp emergent spikes at every square index, and the sequence is provably not a permutation of the positive integers.

Taha Tanver Athar, Tanver Athar42 reads
Preprint · Sep 2026

On the graph-theoretic, spectral, and topological analysis of the Seven Bridges of Königsberg

A formal synthesis of the Seven Bridges of Königsberg, Euler's founding result of 1736, carried into modern language: adjacency, incidence, and Laplacian matrix representations give a self-contained proof of impossibility; the full Laplacian spectrum and eigenbasis, Kirchhoff network flows, and the Kirchhoff index quantify structural efficiency; and two bridge additions — the historical Kaiserbrücke of 1905 and a provably minimal two-bridge Eulerisation — are analyzed exactly.

Taha Athar, Tanver Athar61 reads