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Register a new userLarge prime factors on short intervals
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Jori Merikoski
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We show that for all large enough $x$ the interval $[x,x+x^{1/2}log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals $[x,x+x^{1/2+epsilon}].$ We also incorporate some ideas from Harmans book `Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomaki and Radziwi{l}{l}(2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of $log x$ when applying Harmans sieve method.
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Stewart (2013) proved that the biggest prime divisor of the $n$th term of a Lucas sequence of integers grows quicker than $n$, answering famous questions of ErdH{o}s and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewarts result.
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We call an odd positive integer $n$ a $textit{Descartes number}$ if there exist positive integers $k,m$ such that $n = km$ and begin{equation} sigma(k)(m+1) = 2km end{equation} Currently, $mathcal{D} = 3^{2}7^{2}11^{2}13^{2}22021$ is the only known Descartes number. In $2008$, Banks et al. proved that $mathcal{D}$ is the only cube-free Descartes number with fewer than seven distinct prime factors. In the present paper, we extend the methods of Banks et al. to show that there is no cube-free Descartes number with seven distinct prime factors.
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