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SUMMARY:Wayne Lewis (University of Hawai`i\, Honolulu Community College)
DTSTART:20251219T160000Z
DTEND:20251219T173000Z
DTSTAMP:20260404T094340Z
UID:APRC/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/APRC/
 7/">Infinitude 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 a
 n ultraproduct proof that there are infinitely many Sophie Germain primes 
 (i.e.\, infinitely many primes $p$ such that $(p-1)/2$ is also prime):\n\n
 Complements of the prime sets associated to cyclotomic polynomials have th
 e finite intersection property (by CRT and Dirichlet)\, hence extend to a 
 nonprincipal ultrafilter $\\mathcal{U}$ containing all these complements.\
 nThis yields a characteristic $0$ Henselian valued field $\\widetilde{\\ma
 thbb{Q}}=\\prod_{\\mathcal{U}}\\mathbb{Q}_p$ with valuation domain\n$$\n\\
 widetilde{\\mathbb{Z}}=\\prod_{\\mathcal{U}}\\mathbb{Z}_p=\\mathbb{F}\\opl
 us \\tilde s\\\,\\widetilde{\\mathbb{Z}}\,\n\\qquad\n\\tilde s=(2\,3\,5\,7
 \,11\,\\dots)/\\mathcal{U}\,\n$$\nfor a maximal discrete subfield $\\mathb
 b{F}$.\nSet\n$$\n\\mathbb{L}=\\mathrm{Abs}(\\widetilde{\\mathbb{Q}})=\\big
 cap\\{K:K\\text{ is a maximal discrete subfield of }\\widetilde{\\mathbb{Q
 }}\\}.\n$$\nThen $\\mathrm{tor}(\\mathbb{L}^\\times)=\\{\\pm 1\\}$: one sh
 ows $\\tilde n\\mid(\\tilde s-\\tilde 1)$ if and only if the cyclotomic po
 lynomial $\\Phi_n$ has a zero in $\\widetilde{\\mathbb{Q}}$\, and by const
 ruction of $\\mathcal{U}$ no $\\Phi_n$ with $n>2$ has a zero.\n\nLet $\\wi
 detilde{\\mathbb{B}}\\cong \\mathbb{Z}^{\\mathbb{P}}/\\mathcal{U}$ be the 
 associated Bézout domain with additive group order-isomorphic to the valu
 e group of $\\widetilde{\\mathbb{Q}}$\, and set $\\tilde v=(\\tilde s-\\ti
 lde 1)/\\tilde 2\\in \\widetilde{\\mathbb{B}}$.\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$$\n-\\ti
 lde 1=\\eta(\\tilde v)=\\eta(\\tilde b\\tilde c)=\\eta(\\tilde b)^{\\tilde
  c}.\n$$\nThe internal product decomposition $\\widetilde{\\mathbb{Z}}^\\t
 imes=\\widetilde{\\mu}\\\,(\\tilde 1+\\tilde s\\\,\\widetilde{\\mathbb{Z}}
 )$ forces $\\eta(\\tilde b)\\in\\widetilde{\\mu}$\, so $\\widetilde{\\mu}\
 \cap \\mathrm{tor}(\\mathbb{L}^\\times)=\\{\\pm 1\\}$ gives $\\eta(\\tilde
  b)=-\\tilde 1$ with $\\tilde b<\\tilde v$.\nThis contradicts $[\\tilde 0\
 ,\\tilde s-\\tilde 1)_{\\widetilde{\\mathbb{B}}}$ is a transversal for $\\
 widetilde{\\mathbb{B}}/(\\tilde s-\\tilde 1)\\widetilde{\\mathbb{B}}\\cong
 \\mathbb{F}^\\times$ via $\\eta$.\nHence\, $\\tilde v$ is irreducible in $
 \\widetilde{\\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$$\n\\{p\\in\\mathbb{P}\\colon\\math
 bb{Z}/v_p\\mathbb{Z}\\text{ is a field}\\}\\in\\mathcal{U}\,\n$$\nso $\\{p
 \\in\\mathbb{P}: v_p=(p-1)/2\\text{ is prime}\\}\\in\\mathcal{U}$.\nIn par
 ticular\, infinitely many Sophie Germain primes exist.\n
LOCATION:https://stable.researchseminars.org/talk/APRC/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wayne Lewis (University of Hawai`i\, Honolulu Community College)
DTSTART:20260124T160000Z
DTEND:20260124T173000Z
DTSTAMP:20260404T094340Z
UID:APRC/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/APRC/
 8/">No Spectral Leakage: The Core Mechanism of Quant APRC.</a>\nby Wayne L
 ewis (University of Hawai`i\, Honolulu Community College) as part of Ultra
 products in Number Theory\n\nLecture held in Haiku.\n\nAbstract\nOur guidi
 ng principle is $\\textit{no spectral leakage into the Chebotarev tail}$.\
 nFinite Chebotarev cylinder data already determine the global state—ther
 e is no hidden mass at infinity.\nIn this lecture I state the compatibilit
 y input (Lemma 7.10) and give a detailed proof of the hinge step (Prop. 7.
 11): a projective family of Chebotarev cylinder marginals extends to a uni
 que normal state\, hence $\\omega_\\delta(f_m)=\\lim_{R\\to\\infty}\\omega
 _\\delta(f_{m\,R})$ for $f_{m\,R}\\downarrow f_m$.\n
LOCATION:https://stable.researchseminars.org/talk/APRC/8/
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