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SUMMARY:Wayne Lewis (University of Hawai`i\, Honolulu Community College)
DTSTART:20251219T160000Z
DTEND:20251219T173000Z
DTSTAMP:20260423T003258Z
UID:APRC/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/APRC/7/">Inf
 initude of Sophie Germain Primes</a>\nby Wayne Lewis (University of Hawai`
 i\, Honolulu Community College) as part of Ultraproducts in Number Theory\
 n\nLecture held in Haiku.\n\nAbstract\nWe describe the details of an ultra
 product proof that there are infinitely many Sophie Germain primes (i.e.\,
  infinitely many primes $p$ such that $(p-1)/2$ is also prime):\n\nComplem
 ents of the prime sets associated to cyclotomic polynomials have the finit
 e intersection property (by CRT and Dirichlet)\, hence extend to a nonprin
 cipal ultrafilter $\\mathcal{U}$ containing all these complements.\nThis y
 ields a characteristic $0$ Henselian valued field $\\widetilde{\\mathbb{Q}
 }=\\prod_{\\mathcal{U}}\\mathbb{Q}_p$ with valuation domain\n$$\n\\widetil
 de{\\mathbb{Z}}=\\prod_{\\mathcal{U}}\\mathbb{Z}_p=\\mathbb{F}\\oplus \\ti
 lde s\\\,\\widetilde{\\mathbb{Z}}\,\n\\qquad\n\\tilde s=(2\,3\,5\,7\,11\,\
 \dots)/\\mathcal{U}\,\n$$\nfor a maximal discrete subfield $\\mathbb{F}$.\
 nSet\n$$\n\\mathbb{L}=\\mathrm{Abs}(\\widetilde{\\mathbb{Q}})=\\bigcap\\{K
 :K\\text{ is a maximal discrete subfield of }\\widetilde{\\mathbb{Q}}\\}.\
 n$$\nThen $\\mathrm{tor}(\\mathbb{L}^\\times)=\\{\\pm 1\\}$: one shows $\\
 tilde n\\mid(\\tilde s-\\tilde 1)$ if and only if the cyclotomic polynomia
 l $\\Phi_n$ has a zero in $\\widetilde{\\mathbb{Q}}$\, and by construction
  of $\\mathcal{U}$ no $\\Phi_n$ with $n>2$ has a zero.\n\nLet $\\widetilde
 {\\mathbb{B}}\\cong \\mathbb{Z}^{\\mathbb{P}}/\\mathcal{U}$ be the associa
 ted Bézout domain with additive group order-isomorphic to the value group
  of $\\widetilde{\\mathbb{Q}}$\, and set $\\tilde v=(\\tilde s-\\tilde 1)/
 \\tilde 2\\in \\widetilde{\\mathbb{B}}$.\nIf $\\tilde 1<\\tilde b\,\\tilde
  c<\\tilde v$ with $\\tilde b\\tilde c=\\tilde v$ in $\\widetilde{\\mathbb
 {B}}$\, then for the Kaplansky character $\\eta$ one has\n$$\n-\\tilde 1=\
 \eta(\\tilde v)=\\eta(\\tilde b\\tilde c)=\\eta(\\tilde b)^{\\tilde c}.\n$
 $\nThe internal product decomposition $\\widetilde{\\mathbb{Z}}^\\times=\\
 widetilde{\\mu}\\\,(\\tilde 1+\\tilde s\\\,\\widetilde{\\mathbb{Z}})$ forc
 es $\\eta(\\tilde b)\\in\\widetilde{\\mu}$\, so $\\widetilde{\\mu}\\cap \\
 mathrm{tor}(\\mathbb{L}^\\times)=\\{\\pm 1\\}$ gives $\\eta(\\tilde b)=-\\
 tilde 1$ with $\\tilde b<\\tilde v$.\nThis contradicts $[\\tilde 0\,\\tild
 e s-\\tilde 1)_{\\widetilde{\\mathbb{B}}}$ is a transversal for $\\widetil
 de{\\mathbb{B}}/(\\tilde s-\\tilde 1)\\widetilde{\\mathbb{B}}\\cong\\mathb
 b{F}^\\times$ via $\\eta$.\nHence\, $\\tilde v$ is irreducible in $\\widet
 ilde{\\mathbb{B}}$ and so prime.\n\nFinally\, "is a field" is first-order 
 in the language of rings and\n$$\n\\widetilde{\\mathbb{B}}/\\tilde v\\\,\\
 widetilde{\\mathbb{B}}\\cong \\prod_{\\mathcal{U}}\\mathbb{Z}/v_p\\mathbb{
 Z}.\n$$\nBy Łoś's theorem\,\n$$\n\\{p\\in\\mathbb{P}\\colon\\mathbb{Z}/v
 _p\\mathbb{Z}\\text{ is a field}\\}\\in\\mathcal{U}\,\n$$\nso $\\{p\\in\\m
 athbb{P}: v_p=(p-1)/2\\text{ is prime}\\}\\in\\mathcal{U}$.\nIn particular
 \, infinitely many Sophie Germain primes exist.\n
LOCATION:https://researchseminars.org/talk/APRC/7/
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