Vandermonde's quintic and multiple decompositions of the number 1318

TL;DR

The paper explores a unique numerical identity involving the number 1318 and its multiple decompositions, linking it to Vandermonde's cyclotomic quintic and conjecturing infinite similar decompositions based on algebraic solutions.

Contribution

It introduces a novel decomposition of 1318 related to Vandermonde's quintic and proposes a conjecture on infinitely many analogous decompositions using algebraic solutions.

Findings

Two distinct decompositions of 1318 as sums of products from a specific set.

Connection of these decompositions to Vandermonde's cyclotomic quintic.

Conjecture of infinitely many similar decompositions involving larger sets.

Abstract

This note records a curious numerical identity: the number 1318, connected with Vandermonde's cyclotomic quintic, may be decomposed in two distinct ways as a sum of products of pairs of numbers taken from the set \{$6, 16, 26, 41$\}, namely $1318 = 6\cdot41 + 16\cdot26 + 16\cdot41 = 6\cdot16 + 6\cdot26 + 26\cdot41$. Based on the existence of radical solutions of certain families of Abelian and generalized Abelian equations, we conjecture the existence of an infinite number of analogous decompositions, involving arbitrarily large sets of numbers.