# Explanation of the Tables

by Kyriakos Kefalas

## Theory and Definitions

##### Chebyshev integers

We call an integer y, a Chebyshev integer of the 1st kind if there exist, $n,&space;x\in&space;\mathbb{N}$, with, $n\geq&space;2,&space;x\geq&space;1$, such that

$y=T_{n}(x)$

where, $T_{n}(x)$, is a Chebyshev polynomial of the 1st kind. We do not include in our definition the case, n = 1, since, $T_{1}\left&space;(&space;x&space;\right&space;)=x$. In this site we maintain a table with the largest prime factors, for each Chebyshev integer sequence, $T_{n}(x),&space;n\geq&space;2$, provided that x, is not itself a Chebyshev integer (see below for more details).

##### Functional equation of Chebyshev polynomials

For every, $n,m,x\in&space;\mathbb{N}$, we have

$T_{n&space;m}\left&space;(&space;x&space;\right&space;)=T_{n}\left&space;(T_{m}\left&space;(&space;x&space;\right&space;)&space;\right&space;)=T\left&space;(&space;y&space;\right&space;)$

where, $y\in&space;\mathbb{N}$. Hence we do not maintain tables of Chebyshev sequences if x, itself is a Chebyshev integer, since these cases are covered by other tables. The sequence of Chebyshev integers, $T_{n}\left&space;(&space;x&space;\right&space;)\leq&space;1000$, is

1, 7, 17, 26, 31, 49, 71, 97, 99, 127, 161, 199, 241, 244, 287, 337, 362, 391, 449, 485, 511, 577, 647, 721, 799, 846, 881, 967, …

##### Principal factor of Chebyshev polynomials

The Chebyshev polynomials can be factored into one or more irreducible polynomials over the integers

$T_{n}\left&space;(&space;x&space;\right&space;)&space;=&space;p_{n_{1}}\left&space;(&space;x&space;\right&space;)\cdots&space;p_{n_{k}}\left&space;(&space;x&space;\right&space;)$

$\sum_{j=1}^{k}&space;n_{j}&space;=&space;n$

Let, $n_{1}. We call the polynomial factor, $p_{n_{k}}\left&space;(&space;x&space;\right&space;)$, the principal factor and

$r_{n}\left&space;(&space;x&space;\right&space;)&space;=&space;\frac{T_{n}\left&space;(&space;x&space;\right&space;)}{p_{n_{k}}\left&space;(&space;x&space;\right&space;)}$

the regular factor of the corresponding Chebyshev polynomial.

Since the polynomial factors have integer coefficients they are integers, $\forall&space;x\in&space;\mathbb{N}$. The principal polynomial factor corresponds to the largest integer factor for every, $x\geq&space;2$, and we call it the principal factor of the Chebyshev integer. Analogous for the regular factors. Notice that the principal integer factors are not necessarily prime. A list of prime principal factors is presented below each factorization table. In order to keep the tables as compact as possible,

The main tables contain the prime factorization of the principal factors only.

The polynomial factorization can be easily performed with computer programs like Mathematica etc. For the corresponding theory see:

1.  Rayes, Mohamed Omar, Vilmar Trevisan, and P. Wang. “Factorization of Chebyshev polynomials.” Kent State University. Kent, OH (1998).
##### Critical Factors

Let, $p_{1}<&space;\cdots&space;<&space;p_{k}$, be the distinct prime factors of an integer y, arranged in natural order. We call, $p_{k-1}$, the critical factor of y. The critical factor is the 2nd largest prime factor and essentially determines the difficulty of the factorization.

## What is contained in the Tables

##### Chebyshev sequences of principal factors

For each x, mentioned above, the main factorization Table A contains the prime factorization of the principal factor of the Chebyshev integers, $T_{n}\left&space;(&space;x&space;\right&space;),&space;n\geq&space;2$. We only show prime factors which have at least 21 digits. If there is no complete prime factorization available we show whatever composite factors of the principal factor are known.

##### Table of Record Chebyshev Factorizations

The Table of Record Chebyshev Factorizations contains record factorizations from all Chebyshev sequences of principal factors, whose critical factors have at least 35 digits. They are ordered according to the critical factor.

##### Contributors credited with the discovery

The Record factorization Table contains also the name of the person who claims to take credit of the factorization and the date we have received the relevant information. We publish any personal information as it is given to us. We have no capability to make any cross checking about whether the mentioned person is actually the real discoverer of the factorization. In case of dispute no name will appear. If we discover a credible publication of the factorization by another person at an earlier date, the entry will be changed accordingly. For more details see **

##### How the tables evolve

New entries are added to the tables as they come in. From time to time, the lowest entries of the Record factorization Table will be removed to our archives.

Created 5.5.2019