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SUMMARY:Paolo Leonetti (Universita ``Luigi Bocconi''\, Milano\, Italy)
DTSTART:20220527T140000Z
DTEND:20220527T142500Z
DTSTAMP:20260423T011438Z
UID:CANT2022/47
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/47/
 ">The G.C.D. of $n$ and the $n$th Fibonacci number</a>\nby Paolo Leonetti 
 (Universita ``Luigi Bocconi''\, Milano\, Italy) as part of Combinatorial a
 nd additive number theory (CANT 2022)\n\n\nAbstract\nLet $(F_n)_{n \\geq 1
 }$ be the sequence of Fibonacci numbers\, defined as usual by $F_1 = F_2 =
  1$ and $F_{n + 2} = F_{n + 1} + F_n$ for all positive integers $n$\; and 
 let $\\mathcal{A}$ be the set of all integers of the form $\\gcd(n\, F_n)$
 \, for some positive integer $n$.\nIn this talk we shall illustrate the fo
 llowing result on $\\mathcal{A}$.\n\n\\noindent\n\\textbf{Theorem.} \\text
 it{For all $x \\geq 2$\, we have\n\\begin{equation*}\n\\#\\mathcal{A}(x) \
 \gg \\frac{x}{\\log x} .\n\\end{equation*}\nOn the other hand\, $\\mathcal
 {A}$ has zero asymptotic density.}\nThe proofs rely on a result of Cubre a
 nd Rouse (PAMS\, 2014) which gives\, for each positive integer $n$\, an ex
 plicit formula for the density of primes $p$ such that $n$ divides the ran
 k of appearance of $p$\, that is\, the smallest positive integer $k$ such 
 that $p$ divides $F_k$.\n
LOCATION:https://researchseminars.org/talk/CANT2022/47/
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