A proof of irrationality of $π$ based on nested radicals with roots of $2$

TL;DR

The paper introduces a novel irrationality proof for $\pi$ based on nested radicals defined by $c_k = \sqrt{2 + c_{k-1}}$ with $c_0=0$, yielding arctangent representations that connect to Machin-like formulas. It derives limit expressions for $\pi$ and constructs an arithmetic framework using $\alpha_k = \left\lfloor 2^{k+1}/\pi \right\rfloor$ and fractional parts $\beta_k$ to argue that $\pi$ cannot be rational, further refining with a parity-based sequence $\gamma_k$. The approach also demonstrates practical rational approximations with digits of $\pi$ captured by the method and provides extensive numerical illustrations. Overall, it contributes a theoretically new irrationality proof and a computational mechanism for high-precision rational approximations of $\pi$ through nested radical structures.

Abstract

In this work, we consider four theorems that can be used to prove the irrationality of $π$. These theorems are related to nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}} $ and $c_0 = 0$. Sample computations showing how the rational approximation tend to $π$ with increasing the integer $k$ are presented.

Theorems & Definitions (5)

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