BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Aradhya Goel (Indian Institute of Technology Kanpur) DTSTART:20260713T133000Z DTEND:20260713T135500Z DTSTAMP:20260710T110740Z UID:CANT2026/2 DESCRIPTION:Title: Sophie Germain primes and the totient of Fibonacci numbers\nby Aradhy a Goel (Indian Institute of Technology Kanpur) as part of Combinatorial an d additive number theory seminar (CANT 2026)\n\nLecture held in Science Ce nter in the CUNY Graduate Center (4th floor).\n\nAbstract\nWe study the se t $S(q)$ of residue classes $r$ modulo the Pisano period $\\pi(q)$ for whi ch $q \\mid \\varphi(F_m)$ for every $m \\equiv r \\pmod{\\pi(q)}$. We pro ve that if $q$ is a Sophie Germain prime and $z(2q+1) \\mid \\pi(q)$\, whe re $z$ denotes the rank of apparition\, then $S(q)$ is a nonempty arithmet ic progression\; for $q > 5$\, its cardinality is odd and $q \\equiv 8 \\p mod{15}$. Conversely\, if a prime $p \\equiv 1 \\pmod{q}$ has $z(p) \\mid \\pi(q)$\, then necessarily $p = 2q+1$\, so $q$ is Sophie Germain. \nWe co njecture that $S(q) \\neq \\emptyset$ forces the existence of such a prime $p$\; this is verified for all $q \\leq 50{\,}000$. Assuming the divisibi lity $z(2q+1) \\mid \\pi(q)$ holds for infinitely many Sophie Germain prim es (verified for approximately $23.9\\%$ of the $669$ Sophie Germain prime s $q \\leq 50{\,}000$)\, the Sophie Germain conjecture implies the existen ce of infinitely many primes $q \\equiv 8 \\pmod{15}$ with $(2q+1) \\mid F _{\\pi(q)}$ -- a purely Fibonacci-theoretic condition. \nThese results gen eralize to arbitrary Lucas sequences $U_n(P\,Q)$ with non-square discrimin ant.\n LOCATION:https://researchseminars.org/talk/CANT2026/2/ END:VEVENT END:VCALENDAR