The art of juggling with two balls or A proof for a modular condition of Lucas numbers

TL;DR

This paper explores counting juggling patterns with one or two balls, linking spherical juggling to Mersenne numbers and providing a proof of a Lucas number congruence for primes.

Contribution

It introduces a novel connection between spherical juggling patterns and associated Mersenne numbers, leading to a proof of a Lucas number modular property.

Findings

Connected spherical juggling to associated Mersenne numbers

Proved Lucas number congruence for primes

Counted juggling patterns with one and two balls

Abstract

In this short note we look at the problem of counting juggling patterns with one ball or two balls with a throw at every occurrence. We will do this for both traditional juggling and for spherical juggling. In the latter case we will show a connection to the "associated Mersenne numbers" (A001350) and so as a result will be able to recover a proof that the $p$th Lucas number is congruent to 1 modulo p when p is a prime.