Uniform estimates for smooth polynomials over finite fields by Ofir Gorodetsky

“…A positive integer $n$ is called $m$-_smooth_ if its largest prime factor has size at most $m$. (Sometimes such numbers are called $m$-_friable_: the word "friable" means "crumbly" and captures the idea that $n$ can be broken up into small pieces.) …”
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A generalization of primitive sets and a conjecture of Erdős by Tsz Ho Chan, Jared Duker Lichtman, Carl Pomerance

“…He starts with the observation that if $a$ is a positive integer and $P(a)$ is the largest prime factor of $a$, then $$\prod_{p< P(a)}(1-1/p)\geq \frac c{\log P(a)}\geq\frac c{\log a}$$ for some absolute constant $c>0$. …”
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Concepts studies for future intracity air transportation systems

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A sieve problem over the Gaussian integers by Schlackow, W, Dr Waldemar Schlackow

“…<p>Our main result is that there are infinitely many primes of the form a² + b² such that a² + 4b² has at most 5 prime factors.</p><p>We prove this by first developing the theory of $L$-functions for Gaussian primes by using standard methods. …”

Bounded length intervals containing two primes and an almost-prime by Maynard, J

“…We show under the same assumption that, for any integer k ⩾ 1, there exist constants D(θ, k) and r(θ, k), such that there are infinitely many integers n for which the interval [n, n + D(θ, k)] contains two primes and k almost-primes, with all of the almost-primes having at most r(θ, k) prime factors. If θ can be taken as large as 0.99, and provided that numbers with 2, 3 or 4 prime factors also have level of distribution 0.99, we show that there are infinitely many integers n such that the interval [n, n + 90] contains two primes and an almost-prime with at most four prime factors. …”

When Is the Number of True Different Permutation Polynomials Equal to 0? by Lucian Trifina, Daniela Tarniceriu

“…In this paper, we have obtained the prime factorization form of positive integers <i>N</i> for which the number of true different fourth- and fifth-degree permutation polynomials (PPs) modulo <i>N</i> is equal to zero. …”
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Extending Pollard class of factorable RSA modulus by Abd Ghafar, Amir Hamzah, Kamel Ariffin, Muhammad Rezal, Asbullah, Muhammad Asyraf

“…We remarked here that both p − 1 and q − 1 do not have small prime factors hence they can be generated without error by RSA libraries in current practice.…”
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Topics in analytic number theory by Maynard, J

“…In Chapter 6 we show that there are infinitely many intervals of bounded length containing two primes and a number with at most 31 prime factors. Moreover, if numbers with up to 4 prime factors have `level of distribution' 0.99, there are infinitely many integers n such that the interval [n,n+90] contains 2 primes and an almost-prime with at most 4 prime factors.…”

Bounded Gaps between Products of Special Primes by Ping Ngai Chung, Shiyu Li

“…In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. …”
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Bounded Gaps between Products of Special Primes by Li, Shiyu, Chung, Ping Ngai

“…In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. …”
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On product sets of arithmetic progressions by Max Wenqiang Xu, Yunkun Zhou

“…It follows that almost all elements of the $n\times n$ multiplication table have approximately $2\log\log n$ prime factors, whereas almost all integers between 1 and $n^2$ have approximately $\log\log(n^2)=\log\log n +\log 2$ prime factors, which shows that the multiplication table has size $o(n^2)$. …”
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Factorization by quantum annealing using superconducting flux qubits implementing a multiplier Hamiltonian by Daisuke Saida, Mutsuo Hidaka, Kentaro Imafuku, Yuki Yamanashi

“…Abstract Prime factorization (P = M × N) is a promising application for quantum computing. …”
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A frequent pattern mining algorithm based on FP-growth without generating tree by Tohidi, Hossein, Ibrahim, Hamidah

“…Our algorithm works based on prime factorization, and is called Frequent Pattern-Prime Factorization (FPPF).…”
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Symmetries of abelian Chern-Simons theories and arithmetic by Diego Delmastro, Jaume Gomis

“…The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern-Simons levels, revealing interesting connections with number theory. …”
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Continued Fractions and Unique Factorization on Digraphs by Giscard, P, Thwaite, S, Jaksch, D

“…We give an algorithm producing the prime factorization of individual walks. We obtain a recursive formula producing the prime factorization of ensembles of walks. …”

Uniform estimates for almost primes over finite fields by Elboim, D, Gorodetsky, O

“…We establish a new asymptotic formula for the number of polynomials of degree n with k prime factors over a finite field Fq. The error term tends to 0 uniformly in n and in q. …”

Sieve weights and their smoothings by Granville, A, Koukoulopoulos, D, Maynard, J

“…When 2k is small compared with A, the level of smoothing, then the main contribution to the moments comes from integers with only large prime factors, as one would hope for in sieve weights. …”

Almost prime triples and Chen's theorem by Heath-Brown, D, Li, X

“…We show that there are infinitely many primes p such that not only does p + 2 have at most two prime factors, but p + 6 also has a bounded number of prime divisors. …”

Bounded length intervals containing two primes and an almost-prime II by Maynard, J

“…We show that there are infinitely many intervals of length at most 108 which contain two primes and a number with at most 31 prime factors.…”

RNA secondary structure factorization in prime tangles by Daniele Marchei, Emanuela Merelli

“…In this paper, we extend the tangle-based model with its minimal prime factorization, useful to analyze patterns that characterize the RNA secondary structure. …”
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