Infinitude of Sophie Germain Primes

Abstract: We describe the details of an ultraproduct proof that there are infinitely many Sophie Germain primes (i.e., infinitely many primes $p$ such that $(p-1)/2$ is also prime):

Complements of the prime sets associated to cyclotomic polynomials have the finite intersection property (by CRT and Dirichlet), hence extend to a nonprincipal ultrafilter $\mathcal{U}$ containing all these complements. This yields a characteristic $0$ Henselian valued field $\widetilde{\mathbb{Q}}=\prod_{\mathcal{U}}\mathbb{Q}_p$ with valuation domain $$ \widetilde{\mathbb{Z}}=\prod_{\mathcal{U}}\mathbb{Z}_p=\mathbb{F}\oplus \tilde s\,\widetilde{\mathbb{Z}}, \qquad \tilde s=(2,3,5,7,11,\dots)/\mathcal{U}, $$ for a maximal discrete subfield $\mathbb{F}$. Set $$ \mathbb{L}=\mathrm{Abs}(\widetilde{\mathbb{Q}})=\bigcap\{K:K\text{ is a maximal discrete subfield of }\widetilde{\mathbb{Q}}\}. $$ Then $\mathrm{tor}(\mathbb{L}^\times)=\{\pm 1\}$: one shows $\tilde n\mid(\tilde s-\tilde 1)$ if and only if the cyclotomic polynomial $\Phi_n$ has a zero in $\widetilde{\mathbb{Q}}$, and by construction of $\mathcal{U}$ no $\Phi_n$ with $n>2$ has a zero.

Let $\widetilde{\mathbb{B}}\cong \mathbb{Z}^{\mathbb{P}}/\mathcal{U}$ be the associated Bézout domain with additive group order-isomorphic to the value group of $\widetilde{\mathbb{Q}}$, and set $\tilde v=(\tilde s-\tilde 1)/\tilde 2\in \widetilde{\mathbb{B}}$. If $\tilde 1<\tilde b,\tilde c<\tilde v$ with $\tilde b\tilde c=\tilde v$ in $\widetilde{\mathbb{B}}$, then for the Kaplansky character $\eta$ one has $$ -\tilde 1=\eta(\tilde v)=\eta(\tilde b\tilde c)=\eta(\tilde b)^{\tilde c}. $$ The internal product decomposition $\widetilde{\mathbb{Z}}^\times=\widetilde{\mu}\,(\tilde 1+\tilde s\,\widetilde{\mathbb{Z}})$ 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 $\tilde b<\tilde v$. This 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 $\eta$. Hence, $\tilde v$ is irreducible in $\widetilde{\mathbb{B}}$ and so prime.

Finally, "is a field" is first-order in the language of rings and $$ \widetilde{\mathbb{B}}/\tilde v\,\widetilde{\mathbb{B}}\cong \prod_{\mathcal{U}}\mathbb{Z}/v_p\mathbb{Z}. $$ By Łoś's theorem, $$ \{p\in\mathbb{P}\colon\mathbb{Z}/v_p\mathbb{Z}\text{ is a field}\}\in\mathcal{U}, $$ so $\{p\in\mathbb{P}: v_p=(p-1)/2\text{ is prime}\}\in\mathcal{U}$. In particular, infinitely many Sophie Germain primes exist.

commutative algebranumber theory

Audience: researchers in the discipline

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Ultraproducts in Number Theory

Series comments: https://hawaii.zoom.us/j/87136740867

Meeting ID: 871 3674 0867 Passcode: 219028

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