Divisors of, \(x={2}^{n}+{3}^{n}+{36}^{n}\)

Kyriakos Kefalas - 6 Oct 2026

Abstract

Very large integers, sometimes of the order of millions of digits, which are impossible to completely factorize via the known general factorization algorithms. Even partial factorizations are very difficult. For these integers we apply special factorization methods to achieve a partial factorization. The interest lies not only in the prime divisors themselves, but in the methods used.

1. Prime divisors of \(x\), for, \(n=543083\)

1.1 Prime divisors

We consider the prime divisors of

with

The next table contains prime divisors of \(x\).

No other prime divisors are known. The large prime divisor is very difficult to find with known general factorization algorithms, due to the tremendous size of \(x\). There remains the large composite factor \(\text{C845177}\), where the number is the number of decimal digits.

1.2 Methods

Trial division

Prime divisors \({p}_{1},\,{p}_{2}\), have been found by trial division. All prime divisors up to, \(p<3\,{10}^{10}\), have been checked.

Cube roots of unity

Prime divisor \({p}_{3}\), has been found by the cube roots of unity method. In particular

No other such prime divisor exists.