Related papers: Deterministic factorization of sums and difference…

The best known unconditional deterministic complexity bound for computing the prime factorization of an integer N is O(M_int(N^(1/4) log N)), where M_int(k) denotes the cost of multiplying k-bit integers. This result is due to…

Building on techniques recently introduced by the second author, and further developed by the first author, we show that a positive integer $N$ may be rigorously and deterministically factored into primes in at most \[ O\left( \frac{N^{1/5}…

The best deterministic unconditionally proven integer factorization algorithms have exponential running time complexities of O(N^(1/4)) arithmetic operations, and conditional on the Riemann hypothesis, there is a deterministic algorithm of…

Hittmeir recently presented a deterministic algorithm that provably computes the prime factorisation of a positive integer $N$ in $N^{2/9+o(1)}$ bit operations. Prior to this breakthrough, the best known complexity bound for this problem…

A deterministic algorithm for factoring $n$ using $n^{1/3+o(1)}$ bit operations is presented. The algorithm tests the divisibility of $n$ by all the integers in a short interval at once, rather than integer by integer as in trial division.…

To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…

Let $a,k\in\mathbb{N}$. For the $k-1$-th iterate of the exponential function $x\mapsto a^x$, also known as tetration, we write \[ ^k a:=a^{a^{.^{.^{.^{a}}}}}. \] In this paper, we show how an efficient algorithm for tetration modulo natural…

A new integer deterministic factorization algorithm, rated at arithmetic operations to $O(N^{1/6+\varepsilon})$ arithmetic operations, is presented in this note. Equivalently, given the least $(\log N)/6$ bits of a factor of the balanced…

This paper presents the concept of digit polynomials, which leads to a deterministic and unconditional integer factorization algorithm with the runtime complexity $\mathcal{O}(N^{1/4+\epsilon})$. Strassen's well known factoring approach is…

We revisit the problem of rigorously and deterministically finding elements of large order in the multiplicative group of integers modulo a natural number $N$. Solving this problem is an essential step in several recent deterministic…

Fermat's well-known factorization algorithm is based on finding a representation of natural numbers $N$ as the difference of squares. In 1895, Lawrence generalized this idea and applied it to multiples $kN$ of the original number. A…

The theoretical aspects of four integer factorization algorithms are discussed in details in this note. The focus is on the performances of these algorithms on the subset of hard to factor balanced integers N = pq, p < q < 2p. The running…

Given a large positive integer $N$, how quickly can one construct a prime number larger than $N$ (or between $N$ and 2N)? Using probabilistic methods, one can obtain a prime number in time at most $\log^{O(1)} N$ with high probability by…

We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm…

In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,...,p_k$, and for fixed integers $m_1,...,m_k$, with $p_i\not|m_i$, the numbers $(e_{p_1}(n),...,e_{p_k}(n))$ are…

Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the…

For any fixed $k\geq 2$, we prove that every sufficiently large integer can be expressed as the sum of a $k$th power of a prime and a number with at most $M(k)=6k$ prime factors. For sufficiently large $k$ we also show that one can take…

We introduce a new deterministic factoring algorithm, which could be described in the cryptographically fashionable term of "factoring with hints": we show that, given the knowledge of the factorisations of $O(N^{1/3+\epsilon})$ terms…

Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an…

We derive new formulas for the number of unordered (distinct) factorizations with $k$ parts of a positive integer $n$ as sums over the partitions of $k$ and an auxiliary function, the number of partitions of the prime exponents of $n$,…