Fun With Very Large Numbers

TL;DR

This paper presents formulas involving the sinc function that hold for extremely large values of N, up to about 10^102832732165, and discusses their failure beyond these bounds, including an example surpassing Skewes numbers.

Contribution

It introduces specific sinc-based formulas valid over enormous ranges of N and analyzes their limitations at astronomically large N values.

Findings

Formulas hold for N up to approximately 10^102832732165.

Another example remains valid until N exceeds a double exponential of e.

The second example surpasses the size of Skewes numbers.

Abstract

We give an example of a formula involving the sinc function that holds for every N = 0, 1, 2, ..., up to about 10^102832732165, then fails for all larger N. We give another example that begins to fail after about N ~ exp(exp(exp(exp(exp(exp(e)))))). This number is larger than the Skewes numbers.