Remarks on AKS Primality Testing Algorithm and A Flaw in the Definition of P

TL;DR

This paper critiques the practicality of the AKS primality testing algorithm, highlighting its enormous storage requirements and discussing a flaw in the traditional understanding of the complexity class P, which impacts the perceived feasibility of polynomial-time algorithms.

Contribution

It reveals a significant flaw in the complexity assumptions of P and discusses the impracticality of AKS due to enormous storage needs for large numbers.

Findings

AKS requires about 1 billion gigabytes for 1024-bit numbers

The complexity class P's definition overlooks data read/write time

Practicality issues challenge the significance of P=NP question

Abstract

We remark that the AKS primality testing algorithm [Annals of Mathematics 160 (2), 2004] needs about 1,000,000,000 G (gigabyte) storage space for a number of 1024 bits. The requirement is very hard to meet. The complexity class P which contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, is generally believed to be ``easy". We point out that the time is estimated only in terms of the amount of arithmetic operations. It does not comprise the time for reading and writing data on the tape in a Turing machine. The flaw makes some deterministic polynomial time algorithms impractical, and humbles the importance of P=NP question.