Retriangular numbers and affine maps between polygonal numbers
Mateo Matijasevick
math.NT
Oct 26, 2026 · v1
TL;DR
All results, including claims in remarks, examples and tables, are formalized in Lean 4 with Mathlib across 90 formal statements, prepared with AI assistance.
Abstract
A positive integer $a$ is called retriangular if there is a positive integer $b$ such that $aT+b$ is a triangular number whenever $T$ is. We show that the retriangular numbers are exactly the odd squares greater than $1$, and that for each of them $b$ is unique. The proof rests on a rigidity principle: a non-constant affine map carrying the values of one quadratic sequence into the values of another is induced by an affine map of the plane sending the vertex of one parabola to the vertex of the other. We use it to classify all affine maps between sets of polygonal numbers; the $s$-gonal numbers map affinely into the $t$-gonal numbers if and only if $M_t\mid M_s$, where $M_s=2(s-2)/\gcd(s,4)$ defines a bijection from $\{3,4,5,\dots\}$ onto the positive integers. We also treat sparse data, polynomial maps, figurate numbers of higher degree and sums of two triangular numbers, and we study the monoid of affine self-maps of an arbitrary set of integers. For the $s$-gonal numbers this monoid is an arithmetical congruence monoid, and the factorization of self-maps into irreducible ones is unique only for triangular and square numbers.
Problem
A positive integer a is retriangular if some positive b makes aT+b triangular for every triangular number T. The paper determines all retriangular numbers and, more generally, all affine maps between sets of polygonal numbers.
Approach
The central tool is a rigidity principle. A non-constant affine map sending the values of one quadratic sequence into the values of another must come from an affine map of the plane that sends vertex to vertex. This principle is applied to s-gonal numbers through their vertex forms. The paper also treats sparse data via generalized Pell equations, polynomial maps, and higher-degree figurate numbers. It studies the monoid of affine self-maps of a set of integers. All results are formalized in Lean 4 with Mathlib.
Results
The retriangular numbers are exactly the odd squares greater than 1, and b is unique for each. The s-gonal numbers map affinely into the t-gonal numbers if and only if M_t divides M_s, where M_s = 2(s-2)/gcd(s,4) is a bijection onto the positive integers. For s-gonal numbers, the self-map monoid is an arithmetical congruence monoid, and its factorization is unique only for triangular and square numbers.
