Summa Summarum: Moessner's Theorem without Dynamic Programming

TL;DR

This paper simplifies Moessner's theorem by removing the need for dynamic programming, leading to a clearer understanding and more efficient computation of integral powers and related additive functions.

Contribution

It introduces a dynamic programming-free approach to Moessner's process, simplifying the theorem and enabling more efficient calculations of powers and factorials.

Findings

Simpler statement of Moessner's theorem

More efficient implementation for computing powers

Abstracted additive computations applicable to factorials

Abstract

Seventy years on, Moessner's theorem and Moessner's process -- i.e., the additive computation of integral powers -- continue to fascinate. They have given rise to a variety of elegant proofs, to an implementation in hardware, to generalizations, and now even to a popular video, "The Moessner Miracle.'' The existence of this video, and even more its title, indicate that while the "what'' of Moessner's process is understood, its "how'' and even more its "why'' are still elusive. And indeed all the proofs of Moessner's theorem involve more complicated concepts than both the theorem and the process. This article identifies that Moessner's process implements an additive function with dynamic programming. A version of this implementation without dynamic programming (1) gives rise to a simpler statement of Moessner's theorem and (2) can be abstracted and then instantiated into related additive computations. The simpler statement also suggests a simpler and more efficient implementation to compute integral powers as well as simple additive functions to compute, e.g., Factorial numbers. It also reveals the source of -- to quote John Conway and Richard Guy -- Moessner's magic.