On Packing Thirteen Points in an Equilateral Triangle

The conversation of how to maximize the minimum distance between points - or, equivalently, pack congruent circles- in an equilateral triangle began by Oler in the 1960s. In a 1993 paper, Melissen proved the optimal placements of 4 through 12 points in an equilateral triangle using only partitions and direct applications of Dirichlet’s pigeon-hole principle. In the same paper, he proposed his conjectured optimal arrangements for 13, 14, 17, and 19 points in an equilateral triangle. In 1997, Payan proved Melissen’s conjecture for the arrangement of fourteen points; and, in September 2020, Joos proved Melissen’s conjecture for the optimal arrangement of thirteen points. These proofs completed the optimal arrangements of up to and including fifteen points in an equilateral triangle. Unlike Melissen’s proofs, however, Joos’s proof for the optimal arrangement of thirteen points in an equilateral triangle requires continuous functions and calculus. I propose that it is possible to continue Melissen’s line of reasoning, and complete an entirely discrete proof of Joos’s Theorem for the optimal arrangement of thirteen points in an equilateral triangle. In this paper, we make progress towards such a proof. We prove discretely that if either of two points is fixed, Joos’s Theorem optimally places the remaining twelve. KEYWORDS: optimization; packing; equilateral triangle; distance; circles; points; thirteen; maximize

Schematic Refutations of Formula Schemata

Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

MULTIPLE PATTERNS OF k-TUPLES OF INTEGERS

The first proposition and its corollary are a transfiguration of Dirichlet's pigeon-hole principle. They are applied to show that a wide variety of sequences display arbitrarily large patterns of sums, differences, higher differences, etc. Among these, we include sequences of primes in arithmetic progressions, of powerful integers, sequences of integers with radical index having a prescribed lower bound, and many others. We also deal with patterns in iterated sequences of primes, patterns of gaps between primes, patterns of values of Euler's φ-function, or their gaps, as well as patterns related to the sequence of Carmichael numbers.