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SUMMARY:Solaleh Bolvardizadeh (University of Lethbridge)
DTSTART:20221121T190000Z
DTEND:20221121T200000Z
DTSTAMP:20260423T035408Z
UID:NTC/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTC/9/">On t
 he Quality of the $ABC$-Solutions</a>\nby Solaleh Bolvardizadeh (Universit
 y of Lethbridge) as part of Lethbridge number theory and combinatorics sem
 inar\n\nLecture held in University of Lethbridge: M1040 (Markin Hall).\n\n
 Abstract\nThe quality of the triplet $(a\,b\,c)$\, where $\\gcd(a\,b\,c) =
  1$\, satisfying $a + b = c$ is defined as\n$$\nq(a\,b\,c) = \\frac{\\max\
 \{\\log |a|\, \\log |b|\, \\log |c|\\}}{\\log \\mathrm{rad}(|abc|)}\,\n$$\
 nwhere $\\mathrm{rad}(|abc|)$ is the product of distinct prime factors of 
 $|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjecture st
 ates that given $\\epsilon > 0$ the number of the $ABC$-solutions $(a\,b\,
 c)$ with $q(a\,b\,c) \\geq 1 + \\epsilon$ is finite.\n\nIn the first part 
 of this talk\, under the $ABC$-conjecture\, we explore the quality of cert
 ain families of the $ABC$-solutions formed by terms in Lucas and associate
 d Lucas sequences. We also introduce\, unconditionally\, a new family of $
 ABC$-solutions that has quality $> 1$.\n\nIn the remaining of the talk\, w
 e prove a conjecture of Erd\\"os on the solutions of the Brocard-Ramanujan
  equation\n$$\nn! + 1 = m^2\n$$\nby assuming an explicit version of the $A
 BC$-conjecture proposed by Baker.\n
LOCATION:https://researchseminars.org/talk/NTC/9/
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