arnav garg

research

papers.

my papers as they stand on arXiv. one of them is about the game Chomp, so you can play its solver below.

A note on partitions in the image of pre₂

arXiv:2606.02683 · combinatorics · june 2026

  • ·Proved that pre₂(n) = 1 if and only if n ∈ {1, 2, 4}, and pre₂(n) ≥ 2 for all n ≥ 5.
  • ·Solves Problem 3 from a paper accepted to the Bulletin of the Australian Mathematical Society.
Structural Conjectures for 4×n Chomp: Unique Extension, Asymptotic Ratios, and Period-112 Geometry

arXiv:2604.25952 · combinatorial game theory · april 2026

  • ·A computational study of P-positions in 4×n Chomp. Tabulates over 961 million P-positions for n ≤ 3000.
  • ·Corrects the original asymptotic conjecture, identifying a bimodal HIGH/LOW structure.
  • ·Proves the Unique Extension property as Lemma 1. Two conjectures, a period-112 modular structure and a linear cone geometry, remain open.
  • ·Contributed the integer sequence A395126 to the OEIS.
Collatz-Type Dynamics on Gaussian Integers

DOI 10.5281/zenodo.19078382 · number theory · march 2026

  • ·Extends the Collatz conjecture to the Gaussian integers under four generalizations.
  • ·Found a stable periodic orbit of length 40, with every element a dyadic Gaussian rational.
  • ·Computed the fractal dimension of stability boundaries across 1,000 orbits.

selected results

pre⁡2(n)=1  ⟺  n∈{1,2,4},pre⁡2(n)≥2    (n≥5)\operatorname{pre}_2(n)=1 \iff n\in\{1,2,4\}, \qquad \operatorname{pre}_2(n)\ge 2 \;\; (n\ge 5)
a note on partitions in the image of pre₂ · arXiv:2606.02683
#{P-positions of 4×n Chomp}  >  9.61×108,n≤3000\#\{\text{P-positions of } 4\times n \text{ Chomp}\} \;>\; 9.61\times 10^{8}, \quad n \le 3000
structural conjectures for 4×n chomp · arXiv:2604.25952
∃ O⊂Z[i]:  T(O)=O,    ∣O∣=40\exists\, O \subset \mathbb{Z}[i] : \; T(O)=O, \;\; |O| = 40
collatz-type dynamics on gaussian integers · zenodo

interactive · partitions

a lot of my work is about integer partitions. type one below to see its young diagram, and its conjugate, the reflection across the main diagonal that a lot of partition identities turn on.

n = 12 · 4 parts

diagram · (5, 3, 3, 1)

conjugate · (4, 3, 3, 1, 1)

interactive · chomp

Chomp is a two player game on a grid where the poisoned corner loses. the first player has a winning strategy on any non trivial board, but no general formula for it is known. this board plays the losing positions out exactly. try to beat it.

4×6 chomp
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solver’s move.

961m p-positions for n up to 3000 · oeis a395126 · arxiv:2604.25952

oeis a395126oeis.org/A395126 →

p-positions in 4×n chomp

a sequence i contributed to the oeis. see the full list there.

security research