Results 11 to 20 of about 542,394 (229)

The median largest prime factor

Journal of Number Theory, 2014
Let $M(x)$ denote the median largest prime factor of the integers in the interval $[1,x]$. We prove that $$M(x)=x^{\frac{1}{\sqrt{e}}\exp(-\text{li}_{f}(x)/x)}+O_ε(x^{\frac{1}{\sqrt{e}}}e^{-c(\log x)^{3/5-ε}})$$ where $\text{li}_{f}(x)=\int_{2}^{x}\frac{\{x/t\}}{\log t}dt$. From this, we obtain the asymptotic $$M(x)=e^{\frac{γ-1}{\sqrt{e}}}x^{\frac{1}{\
Naslund, Eric
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The largest prime factor of X ³ +2 [PDF]

Proceedings of the London Mathematical Society, 2001
Let \(f\) be an irreducible polynomial with positive leading coefficient, and define \(P(x;f)\) to be the largest prime divisor of \(\prod_{n\leq x} f(n)\). \textit{C. Hooley} [J. Reine Angew. Math. 303/304, 21--50 (1978; Zbl 0391.10028)] gave a proof that \(P(x,X^3+2) > x^{31/30}\) provided one assumes ``Hypothesis \(R^*\)'', a best possible estimate ...
Heath-Brown, D. R.
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Distributional properties of the largest prime factor [PDF]

Michigan Mathematical Journal, 2005
Let \(P(n)\) denote the largest prime factor of an integer \(n>1\), and \(P(1)=1\), also let \(\text{ e}(z) = \exp(2\pi iz)\). The authors consider the function \(\rho(x;q,a) = \sum_{n\leq x,P(n)\equiv a \bmod q}1\). In the case of \(q\) fixed, this question has been considered previously by the reviewer [Acta Arith. 71, 229--251 (1995; Zbl 0820.11052)]
Banks, William David, 1964-, Harman, G. (Glyn), 1956-, Shparlinski, Igor E.   +2 more
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On the largest prime factor of numerators of Bernoulli numbers [PDF]

Indagationes Mathematicae, 2012
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Bérczes, Attila, Luca, Florian
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Modeling the health impact of legislation to limit the salt content of bread in Portugal: A macro simulation study

Frontiers in Public Health, 2022
BackgroundExcessive salt consumption—associated with a range of adverse health outcomes—is very high in Portugal, and bread is the second largest source.
Francisco Goiana-da-Silva, Francisco Goiana-da-Silva, David Cruz-e-Silva, Ana Rito, Carla Lopes, Carla Lopes, Magdalena Muc, Ara Darzi, Fernando Araújo, Marisa Miraldo, Alexandre Morais Nunes, Luke N. Allen   +11 more
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On the largest prime factor of $x^{2}-1$ [PDF]

Mathematics of Computation, 2010
In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$ such that all prime factors of each of $x, x+1,..., x+n$ are less than 100.
Luca, Florian, Najman, Filip
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The Effect of Fear of Infection and Sufficient Vaccine Reservation Information on Rapid COVID-19 Vaccination in Japan: Evidence From a Retrospective Twitter Analysis

Journal of Medical Internet Research, 2022
BackgroundThe global public health and socioeconomic impacts of the COVID-19 pandemic have been substantial, rendering herd immunity by COVID-19 vaccination an important factor for protecting people and retrieving the economy ...
Qian Niu, Junyu Liu, Masaya Kato, Momoko Nagai-Tanima, Tomoki Aoyama   +4 more
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ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES [PDF]

Journal of the Australian Mathematical Society, 2007
Denote by$\mathbb{P}$the set of all prime numbers and by$P(n)$the largest prime factor of positive integer$n\geq 1$with the convention$P(1)=1$. In this paper, we prove that, for each$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant$c(\unicode[STIX]{x1D702})>1$such that, for every fixed nonzero integer$a\in \mathbb{Z}^{\
Banks, William D., Shparlinski, Igor E.
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On values taken by the largest prime factor of shifted primes (II) [PDF]

International Journal of Number Theory, 2019
Denote by [Formula: see text] the set of all primes and by [Formula: see text] the largest prime factor of integer [Formula: see text] with the convention [Formula: see text]. Let [Formula: see text] be the unique positive solution of the equation [Formula: see text] in [Formula: see text]. Very recently Wu proved that for [Formula: see text] there is
Chen, Bin, Wu, Jie
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Normal largest gap between prime factors [PDF]

Journal de théorie des nombres de Bordeaux, 2020
Let {p j (n)} j=1 ω(n) denote the increasing sequence of distinct prime factors of an integer n. We provide details for the proof of a statement of Erdős implying that, for any function ξ(n) tending to infinity with n, we havef(n):=max1⩽j<ω(n)loglogpj+1(n)logpj(n)=log3n+O(ξ(n))for almost all integers n.
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