Problems

Let $p_j$ denote the $j$-th prime. Is it true that for every integer $n>13$ there exists a prime $q<n$ such that both $q+2$ and $p_{n-q}+q+1$ are prime?

Let \(p_j\) be the jth prime, with \(p_1=2\). Does every integer \(n>5\) admit an integer \(0<k<n\) such that \(p_k p_{n-k}-6\) is prime?

For each integer \(n\ge1\), define \(A_n(b)=\sum_{k=0}^{n}(2k+1)b^{n-k}\) and \(B_n(b)=\sum_{k=0}^{n}(2k+1)b^k\). Is it true that for every \(n\ge1\) there are infinitely many integers \(b>2n+1\) for which both \(A_n(b)\) and \(B_n(b)\) are prime?

Let \(p_j\) be the jth prime, with \(p_1=2\). For every positive integer \(n\notin\{1,2,3,6\}\), does there exist a prime number \(q<n\) such that \(p_q^2+(p_n-1)^2\) is prime?

Let $X=(X_1,\ldots,X_n)$ with $X_i\stackrel{\mathrm{iid}}{\sim}N(0,1)$. For $\sigma\in\{-1,+1\}^n$ define $D_X(\sigma)=|\langle X,\sigma\rangle|$ and $D_n^\star=\min_{\sigma\in\{-1,+1\}^n}D_X(\sigma)$. Global sign symmetry identifies $\sigma$ and $-\sigma$. Fix $\delta\in(0,1)$. In a uniform randomized exact-real BSS-style model, an algorithm receives the full exact real vector $X$, may perform unit-cost arithmetic and comparisons and generate unbiased random bits, and has worst-case runtime at most $T(n)$ over inputs and random tapes. Define $\operatorname{Opt}_{T,\delta}(n)$ as the infimum over uniform randomized algorithms of the infimum $\varepsilon>0$ such that $\Pr_{X,R}[D_X(A(X;R))\le\varepsilon]\ge1-\delta$. Determine whether, for fixed $\delta$, $\log_2(1/\operatorname{Opt}_{\mathrm{poly},\delta}(n))=\Theta((\log_2 n)^2)$, equivalently whether the best polynomial-time discrepancy is $2^{-\Theta((\log n)^2)}$ up to multiplicative factors with logarithm $o((\log n)^2)$. The broader extension asks for nontrivial bounds on $T$ versus $\operatorname{Opt}_{T,\delta}(n)$ between polynomial and exponential time. Distinguish unconditional results, restricted algorithm-class barriers, conditional lattice hardness, heuristics, and computational observations.

For each integer \(n\geq0\), define \[a_n=\sum_{i=0}^{n}\sum_{j=0}^{n}\binom ni^2\binom nj^2\binom{n+i}{n}\binom{i+j}{i}.\] Is \(n+1\mid a_n\) for every integer \(n\geq0\)?

For each integer \(n\ge2\), let \(f_n(X)=X^n-X-1\). Is it true that for every \(n\ge2\) there is a prime \(p<n(n+3)/2\) for which \(f_n\) is irreducible in \(\mathbb F_p[X]\)?

For each integer \(n\ge2\), define \(F_n(s)=\sum_{k=1}^{n}k s^{k-1}\). Is it true that for every \(n\ge2\) there are infinitely many integers \(s>n\) for which \(F_n(s)\) is prime, and that at least one such integer satisfies \(s<12n^2\)?

For a positive integer \(n=\prod_p p^{e_p}\), define \(s(n)=\sum_p e_pp\), with \(s(1)=0\). Let \(F_0=0,F_1=1\), and \(F_{j+2}=F_{j+1}+F_j\) for \(j\geq0\). Are there infinitely many integers \(n\geq1\) for which both \(s(n)\) and \(s(n+1)\) are Fibonacci numbers?

Let \(p_j\) be the \(j\)-th prime. Is it true that for every integer \(m\geq2\) with \(m\ne8\), there exist an integer \(r\geq1\) and a prime \(q>p_{r+m-2}\) such that \[m^3=\sum_{j=r}^{r+m-2}p_j+q\]?

Let \(p_j\) be the \(j\)-th prime, with \(p_1=2\). Is it true that for every integer \(n\ge2\) there is an integer \(z\) with \(1\le z\le p_{2n-2}\) for which \(z^n-z-1\) is prime?

Let \(p_i\) be the \(i\)-th prime and let \(p_i\#=\prod_{h=1}^i p_h\). Let \(S\) consist of the positive composite integers \(N\) satisfying \(2^{N-1}\equiv1\pmod N\) that have a representation \(N=j(p_i\#)+\varepsilon q\), where \(i\geq1\), \(q\) is prime, \(p_i<q<p_{i+1}^2\), \(1\leq j<p_{i+1}\) is an integer, and \(\varepsilon\in\{-1,1\}\). Is \(S=\{341,1387,2047,4681,13747\}\)?

Define \(T\) on the positive integers by \(T(1)=1\), \(T(x)=2x-1\) when \(x\) is prime, and \(T(x)=\max\{d:1\leq d<x,\ d\mid x\}\) when \(x\) is composite. Is it true that for every integer \(n\geq2\), some iterate \(T^r(n)\), with integer \(r\geq0\), equals 3 or 19?

Given $κ>1$ and an exponential-moment bound on the empirical degree distributions $|λ_k(B)|\geq\sqrtκ-o_N(1)$, where $N$ is the number of vertices.

Let \(\Phi\) be the classical even Riemann theta kernel and define \[ C_\Phi(a,b)=2\int_{|a+b|/2}^{\infty}q\,\Phi\!\left(\frac{a-b}{2}+q\right)\Phi\!\left(\frac{a-b}{2}-q\right)\,dq. \] Fix the interval \([-2,2]\). For each even integer \(m\ge 2\), set \(h=4/m\) and \(t_j=-2+jh\) for \(0\le j\le m\), and form the anchor Gram matrix \(G_m=[C_\Phi(t_i,t_j)]_{0\le i,j\le m}\). Whenever \(G_m\) is invertible, define \[ c_m(a)=\bigl(C_\Phi(t_0,a),\ldots,C_\Phi(t_m,a)\bigr)^T \] and the intrinsic Schur residual \[ S_m(a,b)=C_\Phi(a,b)-c_m(a)^*G_m^{-1}c_m(b). \] For the cell \(I_j=[t_j,t_{j+1}]\), define the normalized Dirichlet modes \[ \phi_{j,k}(x)=\sqrt{\frac2h}\sin\!\left(k\pi\frac{x-t_j}{h}\right),\qquad k\ge1. \] For \(k,\ell\ge1\), let \(H_{k\ell}^{(m)}\) be the \(m\times m\) block whose \((i,j)\) entry is \[ (H_{k\ell}^{(m)})_{ij}=\int_{I_i}\int_{I_j}S_m(x,y)\phi_{i,k}(x)\phi_{j,\ell}(y)\,dx\,dy. \] Whenever \(H_{11}^{(m)}\) is invertible, define \[ P_1^{(m)}=H_{11}^{(m)},\qquad P_2^{(m)}=H_{22}^{(m)}-H_{21}^{(m)}(H_{11}^{(m)})^{-1}H_{12}^{(m)}. \] Determine the exact small-cell asymptotic of the first even modal pivot \(P_2^{(m)}\) as \(m\to\infty\), equivalently \(h\downarrow0\). In particular, decide whether there exist constants \(m_0\) and \(0<c_-\le c_+<\infty\) such that for every even \(m\ge m_0\), the matrices \(G_m\) and \(H_{11}^{(m)}\) are invertible, the pivots \(P_1^{(m)}\) and \(P_2^{(m)}\) are positive semidefinite, and \[ c_-h^6\le \frac{\operatorname{Tr}P_2^{(m)}}{\operatorname{Tr}P_1^{(m)}}\le c_+h^6. \] Preferably determine whether the finite positive limit \[ \lim_{m\to\infty}h^{-6}\frac{\operatorname{Tr}P_2^{(m)}}{\operatorname{Tr}P_1^{(m)}} \] exists and identify it. A negative resolution may instead exhibit a rigorous infinite sequence of even \(m\to\infty\) for which the required invertibility, positivity, or either side of the \(h^6\) comparison fails.

Let \(\Phi\) be the classical even Riemann theta kernel appearing in the Fourier representation \[ \Xi(z)=2\int_0^\infty \Phi(t)\cos(zt)\,dt. \] For real \(a,b\), define \[ C_\Phi(a,b)=2\int_{|a+b|/2}^{\infty}q\,\Phi\!\left(\frac{a-b}{2}+q\right)\Phi\!\left(\frac{a-b}{2}-q\right)\,dq. \] Write \(D(a)=C_\Phi(a,a)\), and define the normalized kernel \[ R_\Phi(a,b)=\frac{C_\Phi(a,b)}{\sqrt{D(a)D(b)}}. \] The diagonal quantity \(D(a)\) is positive. Fix the Gaussian function \[ u(a)=e^{-3a^2/10}. \] Among all nonnegative real numbers \(\beta\) for which the kernel \[ R_\Phi(a,b)-\beta\,u(a)u(b) \] is positive semidefinite on \(\mathbb R\), let \(\beta_*\) denote the supremum; if there is no positive admissible \(\beta\), set \(\beta_*=0\). Determine whether \[ \beta_*\ge \frac{19}{20}. \] Equivalently, prove or disprove that for every integer \(n\ge1\) and every choice of real numbers \(a_1,\dots,a_n\), the matrix \[ \left[R_\Phi(a_i,a_j)-\frac{19}{20}e^{-3a_i^2/10}e^{-3a_j^2/10}\right]_{i,j=1}^n \] is positive semidefinite. A positive solution must come directly from the theta-side structure above. It may not assume the Riemann hypothesis, the reality of the zeros of \(\Xi\), Hermite--Biehler or de Branges positivity, the Schur quotient \((\Xi-i\Xi')/(\Xi+i\Xi')\), or a pre-existing positivity theorem for \(C_\Phi\) or \(R_\Phi\). A negative solution may consist of an explicit finite configuration, a local jet obstruction, an asymptotic obstruction, or another rigorous certificate. A stronger solution may determine \(\beta_*\) exactly or establish rigorous bounds that decide the stated inequality.

Let \(\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) and let \(\Xi(z)=\xi(\tfrac12+iz)\), so \(\Xi\) is a real even entire function. Let \(\Phi\) denote the standard even Riemann theta kernel normalized by \[ \Xi(z)=2\int_0^\infty \Phi(u)\cos(zu)\,du, \qquad \Phi(u)=\sum_{n\ge1}\bigl(2\pi^2n^4e^{9u/2}-3\pi n^2e^{5u/2}\bigr)e^{-\pi n^2e^{2u}}. \] For \(z,w\) in the upper half-plane \(\mathbb H=\{z:\operatorname{Im}z>0\}\), define the Hermitian Pick/Bezout kernel \[ B_\Xi(z,w)=\frac{\Xi(z)\overline{\Xi'(w)}-\Xi'(z)\overline{\Xi(w)}}{z-\overline w}. \] Construct, directly from \(\Phi\) and explicit integral transforms, an unconditional positive-kernel representation of the form \[ B_\Xi(z,w)=\int_\Omega \Psi_z(\omega)\overline{\Psi_w(\omega)}\,d\mu(\omega),\qquad d\mu\ge0, \] valid for all \(z,w\in\mathbb H\), with convergence and Hermitian analyticity justified. The construction must not assume the Riemann Hypothesis, zero-freeness of \(\Xi\) in \(\mathbb H\), Hermite--Biehler/de Branges positivity, or the Pick positivity being proved. Prove as part of the solution that such a factorization implies that \(-\Xi'/\Xi\) is a Herglotz/Nevanlinna function on its zero-free domain and forces all zeros of \(\Xi\) to be real, hence proves the Riemann Hypothesis. A valid negative resolution may instead prove an obstruction theorem for a mathematically natural, explicitly specified class of theta-side factorizations (for example theta-local, finite-order, tensor-product, or finitely generated transforms), showing that no representation in that class can yield the required global positive kernel. Mere failure of a particular ansatz or finite numerical testing does not count.

Let n,k∈N_{>0}. For z∈C define E^(0)(z)=z and E^(j)(z)=exp(E^(j-1)(z)) for j≥1. Suppose z_1,...,z_n∈C are linearly independent over Q. Prove that \[ \operatorname{trdeg}_{\mathbb Q}\mathbb Q\!\left(\bigcup_{j=0}^{k}\{E^{(j)}(z_1),\ldots,E^{(j)}(z_n)\}\right)\ge kn-(k-1)\operatorname{trdeg}_{\mathbb Q}\mathbb Q(z_1,\ldots,z_n). \] The field on the left is exactly the field generated over Q by all displayed iterated exponentials through level k; no additional constants or base fields are adjoined.

Let \(\lambda\) be the Gaussian angular Hecke character on the nonzero integral ideals of \(\mathbb Z[i]\), defined on principal ideals by \(\lambda((z))=(z/|z|)^4\), and put \(L_k(s)=L(s,\lambda^k)\) for \(k\ge 1\). Fix \(\Phi\in C_c^\infty((1,2))\), a real number \(0<\vartheta<1/2\), and set \(X=K^\vartheta\). Fix a compact set \(T\subset\mathbb R\) and a sufficiently large integer \(J\). For each \(K\), let \(F_K\in C_c^\infty((0,1))\) satisfy \[ \sup_K\sum_{j=0}^{J}\int_{\mathbb R}(1+|\xi|)^J\left|\partial_\xi^j\widehat F_K(\xi)\right|\,d\xi<\infty, \] where \(\widehat F_K(\xi)=\int_{\mathbb R}F_K(u)e^{-iu\xi}\,du\). For \(t_0\in T\), define \[ a_{\mathfrak a}=(N\mathfrak a)^{-1/2-it_0}F_K\!\left(\frac{\log N\mathfrak a}{\log X}\right),\qquad A_k(a)=\sum_{\mathfrak a}a_{\mathfrak a}\lambda(\mathfrak a)^k, \] and \(\|a\|_2^2=\sum_{\mathfrak a}|a_{\mathfrak a}|^2\). Define \[ \mathcal M_K(\alpha,\beta;a)=\sum_{k\ge1}\Phi(k/K)L_k(1/2+\alpha)L_k(1/2+\beta)|A_k(a)|^2. \] Put \(L_0(s)=\zeta(s)L(s,\chi_{-4})\) and \[ Z(\gamma,\delta)=\frac{\zeta(1+2\gamma)\zeta(1+2\delta)L_0(1+\gamma+\delta)}{\zeta(2+2\gamma+2\delta)}. \] For integral ideals \(\mathfrak n\), define \[ \tau_{\gamma,\delta}(\mathfrak n)=\sum_{\mathfrak r\mathfrak s=\mathfrak n}N\mathfrak r^{-\gamma}N\mathfrak s^{-\delta}. \] For a nonzero ideal \(\mathfrak l\), write \(\mathfrak l=(q_{\mathfrak l})\mathfrak l^\circ\), where \((q_{\mathfrak l})\) is the largest ideal generated by a positive rational integer dividing \(\mathfrak l\). If \(\mathfrak p^d\Vert\mathfrak l^\circ\), define \[ P_{\mathfrak p^d}(\gamma,\delta)=\frac{N\mathfrak p^{-d/2}}{1+N\mathfrak p^{-1-\gamma-\delta}}\left(\tau_{\gamma,\delta}(\mathfrak p^d)-N\mathfrak p^{-1-2\gamma-2\delta}\tau_{\gamma,\delta}(\mathfrak p^{d-2})\right), \] with \(\tau_{\gamma,\delta}(\mathfrak p^j)=0\) for \(j<0\), and set \[ P_{\mathfrak l}(\gamma,\delta)=\prod_{\mathfrak p^d\Vert\mathfrak l^\circ}P_{\mathfrak p^d}(\gamma,\delta). \] This convention includes the ramified prime above \(2\); rational ideal content does not change the angular twist. For \(s\in\mathbb C\), define \[ X_k(s)=\pi^{2s}\frac{\Gamma(2k+\tfrac12-s)}{\Gamma(2k+\tfrac12+s)} \] and, for \(\epsilon_1,\epsilon_2\in\{\pm1\}\), \[ G_{\epsilon_1,\epsilon_2}(k;\alpha,\beta)=X_k(\alpha)^{(1-\epsilon_1)/2}X_k(\beta)^{(1-\epsilon_2)/2}. \] Define the exact-gamma four-swap main term by \[ \begin{aligned} \mathcal M_K^{(4),\mathrm{exact}}(\alpha,\beta;a)=&\sum_{\epsilon_1,\epsilon_2=\pm1}\sum_{k\ge1}\Phi(k/K)G_{\epsilon_1,\epsilon_2}(k;\alpha,\beta)\\ &\times\sum_{\mathfrak a,\mathfrak b}a_{\mathfrak a}\overline{a_{\mathfrak b}}\,Z(\epsilon_1\alpha,\epsilon_2\beta)P_{\mathfrak a\overline{\mathfrak b}}(\epsilon_1\alpha,\epsilon_2\beta). \end{aligned} \] The combined four-sector expression is interpreted by common meromorphic continuation whenever separated terms have removable singularities. Define \[ H_k(\alpha,\beta)=\pi^{\alpha+\beta}\frac{\Gamma(2k+\tfrac12)^2}{\Gamma(2k+\tfrac12+\alpha)\Gamma(2k+\tfrac12+\beta)}. \] Call a compact set \(\Omega\subset\mathbb C^2\) regular if the combined four-sector meromorphic expression has no nonremovable pole on \(\Omega\). The collision divisors \(\alpha=0\), \(\beta=0\), \(\alpha+\beta=0\), and \(\alpha-\beta=0\) may be included when the combined expression has a removable continuation there. Prove that for every fixed regular compact \(\Omega\), every \(\varepsilon>0\), and every fixed \(0<\vartheta<1/2\), uniformly for \((\alpha,\beta)\in\Omega\), \(t_0\in T\), and all packets \(F_K\) satisfying the stated norm condition, \[ \mathcal M_K(\alpha,\beta;a)=\mathcal M_K^{(4),\mathrm{exact}}(\alpha,\beta;a)+O_{\Omega,\vartheta,T,\Phi,J,\varepsilon}\!\left(K^{1/2+\varepsilon}\sup_{k\asymp K}|H_k(\alpha,\beta)|\,\|a\|_2^2\right), \] with no hidden factor \(X^\delta\) for any fixed \(\delta>0\). Equivalently, by uniform Stirling asymptotics on every fixed regular compact \(\Omega\), \[ \mathcal M_K=\mathcal M_K^{(4),\mathrm{exact}}+O_{\Omega,\vartheta,T,\Phi,J,\varepsilon}\!\left(K^{1/2-\Re(\alpha+\beta)+\varepsilon}\|a\|_2^2\right). \] The endpoint \(\vartheta=1/2\) and shift ranges growing with \(K\) are excluded.

Let \(K\to\infty\). Let \(\lambda\) be the Gaussian angular Hecke character on the nonzero ideals of \(\mathbb Z[i]\), defined by \(\lambda((z))=(z/|z|)^4\), and let \(L_k(s)=L(s,\lambda^k)\). Fix a smooth weight \(\Phi\) supported in \((1,2)\), a real number \(\vartheta\) with \(0<\vartheta<1/2\), and put \(X=K^\vartheta\). Fix also a compact set \(T\subset\mathbb R\), a compact regular set \(\Omega\) of pairs of shifts \((\alpha,\beta)\), and a sufficiently large fixed integer \(J\). For every \(K\), let \(F_K\in C_c^\infty((0,1))\) satisfy \[ \int_{\mathbb R}(1+|\xi|)^J|\widehat F_K(\xi)|\,d\xi\le 1. \] For \(t_0\in T\), define \[ a_{\mathfrak a}=(N\mathfrak a)^{-1/2-it_0}F_K\!\left(\frac{\log N\mathfrak a}{\log X}\right),\qquad A_k=\sum_{N\mathfrak a\le X}a_{\mathfrak a}\lambda(\mathfrak a)^k, \] where the sums are over nonzero integral ideals of \(\mathbb Z[i]\). Consider the amplified shifted second moment \[ \mathcal M_K(\alpha,\beta;a)=\sum_{k\ge1}\Phi(k/K)|A_k|^2L_k(1/2+\alpha)L_k(1/2+\beta). \] For a nonzero ideal \(\mathfrak l\), define its fixed-twist four-swap main term as follows. Let \(\widetilde\Phi(s)=\int_1^2x^s\Phi(x)\,dx\), let \(L_0(s)=\zeta(s)L(s,\chi_{-4})\), and put \[ Z(\gamma,\delta)=\frac{\zeta(1+2\gamma)\zeta(1+2\delta)L_0(1+\gamma+\delta)}{\zeta(2+2\gamma+2\delta)}. \] Write \(\mathfrak l=(q_{\mathfrak l})\mathfrak l^\circ\), where \((q_{\mathfrak l})\) is the largest ideal generated by a positive rational integer that divides \(\mathfrak l\). If \(\mathfrak p^d\Vert\mathfrak l^\circ\), define \[ \tau_{\gamma,\delta}(\mathfrak n)=\sum_{\mathfrak r\mathfrak s=\mathfrak n}N\mathfrak r^{-\gamma}N\mathfrak s^{-\delta}, \] \[ P_{\mathfrak p^d}(\gamma,\delta)=\frac{N\mathfrak p^{-d/2}}{1+N\mathfrak p^{-1-\gamma-\delta}}\left(\tau_{\gamma,\delta}(\mathfrak p^d)-N\mathfrak p^{-1-2\gamma-2\delta}\tau_{\gamma,\delta}(\mathfrak p^{d-2})\right), \] with \(\tau_{\gamma,\delta}(\mathfrak p^j)=0\) for \(j<0\), and set \(P_{\mathfrak l}(\gamma,\delta)=\prod_{\mathfrak p^d\Vert\mathfrak l^\circ}P_{\mathfrak p^d}(\gamma,\delta)\). Then define \[ \mathcal M_{\mathfrak l}^{(4)}(\alpha,\beta)=K\sum_{\epsilon_1,\epsilon_2\in\{\pm1\}}\left(\frac{2K}{\pi}\right)^{-(1-\epsilon_1)\alpha-(1-\epsilon_2)\beta}\widetilde\Phi\bigl(-(1-\epsilon_1)\alpha-(1-\epsilon_2)\beta\bigr)Z(\epsilon_1\alpha,\epsilon_2\beta)P_{\mathfrak l}(\epsilon_1\alpha,\epsilon_2\beta), \] with the combined expression interpreted by meromorphic continuation when separated terms have removable singularities. Finally put \[ \mathcal M_K^{(4)}(\alpha,\beta;a)=\sum_{\mathfrak a,\mathfrak b}a_{\mathfrak a}\overline{a_{\mathfrak b}}\,\mathcal M_{\mathfrak a\overline{\mathfrak b}}^{(4)}(\alpha,\beta). \] Prove that for every \(\varepsilon>0\), uniformly for \((\alpha,\beta)\in\Omega\), \(t_0\in T\), and every packet \(F_K\) satisfying the stated norm bound, \[ \mathcal M_K(\alpha,\beta;a)=\mathcal M_K^{(4)}(\alpha,\beta;a)+O_{\vartheta,\Omega,T,\Phi,J,\varepsilon}\bigl(K^{1/2+\varepsilon}\|a\|_2^2\bigr). \] The remainder is required to have no hidden factor \(X^\delta\) with fixed \(\delta>0\). The endpoint \(\vartheta=1/2\) and shift ranges growing with \(K\) are not part of the base problem.

For a nonzero lattice point \(v=(m,n)\in\mathbb Z^{2}\), let \(r_{v}=(m^{2}+n^{2})^{1/2}\) and let \(\theta_{v}\) denote the argument of \(m+in\). In the half-plane \(\operatorname{Re}(s)>1\), put \[ C_{0}(s)=\sum_{v\neq(0,0)}r_{v}^{-2s}=4\zeta(s)\beta(s) \] and, for each positive integer \(k\), put \[ C_{k}(s)=\sum_{v\neq(0,0)}\frac{\cos(4k\theta_{v})}{r_{v}^{2s}}. \] Use the meromorphic continuations of these functions and define \[ D_{k}(s)=\frac{C_{0}(s)-C_{k}(s)}{\zeta(2s)}. \] In the half-plane of absolute convergence this is also \[ D_{k}(s)=\sum_{\substack{(m,n)\in\mathbb Z^{2}\\ \operatorname{gcd}(m,n)=1}}\frac{1-\cos(4k\theta_{(m,n)})}{(m^{2}+n^{2})^{s}}. \] For every positive integer \(k\), the function \(D_{k}\) has an unavoidable geometric simple pole at \(s=1\), with residue \(6/\pi\). Remove it by defining \[ \widetilde{D}_{k}(s)=D_{k}(s)-\frac{6}{\pi(s-1)}. \] A separate detection theorem is taken as background: for every nontrivial zero \(\rho\) of the Riemann zeta function, at least one positive integer \(k\) satisfies \(C_{k}(\rho/2)\neq C_{0}(\rho/2)\). Thus any zero with real part greater than \(1/2\) would produce a pole of at least one \(D_{k}\) at \(s=\rho/2\). Prove an independent geometric, analytic, or operator-theoretic pole-exclusion principle, arising from primitive Pythagorean geometry or from an explicitly constructed associated operator system, which implies that \(\widetilde{D}_{k}\) is holomorphic throughout the half-plane \(\operatorname{Re}(s)>1/4\) for every positive integer \(k\). Equivalently for the intended application, it is enough to prove that if \(s_{0}\) lies in that half-plane and \(\zeta(2s_{0})=0\), then \(D_{k}\) is holomorphic at \(s_{0}\) for every positive integer \(k\). The proof may not assume the Riemann hypothesis, the required zero-free region, a Möbius-summatory estimate or reciprocal-zeta estimate already strong enough to imply that region, or global holomorphy of all primitive responses. Any positivity, passivity, contractivity, Hermite–Biehler or de Branges property, Fredholm invertibility, logarithmic factorization, finite-negative-index Pontryagin structure, generalized Schur or Nevanlinna property, coercive estimate, or comparable stability hypothesis used in the proof must itself be derived from an explicit construction without encoding the zeta-zero divisor or an equivalent zero-free assumption. An index or winding calculation must genuinely exclude a local pole rather than merely count a pole together with compensating zeros. A Pontryagin or Krein–Langer argument must begin from an independently constructed finite-index realization with an a priori index bound. A Weil-positivity or Connes–Consani–Moscovici argument may not assume global positivity and must rigorously avoid or overcome the known negative Shannon-transition directions. Regularity, boundedness, smoothness, or decay of a residue or principal part does not by itself exclude a pole. The argument must cover zeros of arbitrary multiplicity. Finite numerical verification may be supporting evidence but cannot satisfy the universal conclusion.

Fix an acceptable universal Turing-machine model \(U\) and a Blum complexity measure \(\Phi\). Fix a sound, recursively axiomatized proof system \(T\) with a recursive proof checker for the formal language of \(U\)-programs, totality, correctness, and asymptotic complexity bounds. A discovery instance is \(I=(e,K,\Delta)\), where \(e\) is a formal index for a total computable decision problem \(L_e\subseteq\{0,1\}^*\), \(K=(k_1,\ldots,k_m)\) is a finite nonempty list of indices for total deciders of \(L_e\), and \(\Delta:\mathbb N\to\mathbb N\) is a total computable target-improvement function. Define \(F_K(n)=\min_i\Phi_{k_i}(n)\). A certified witness is \((a,\pi)\), where \(a\) indexes a total decider \(A\) for \(L_e\) and \(\pi\) is a finite \(T\)-proof of the sentence asserting that \(A\) is total and decides \(L_e\), and that there exists \(N\) such that for every \(n\ge N\), \(\Phi_A(n)\leq\Delta(F_K(n))\). Let \(\mathrm{DISC}_T\) be the set of valid discovery instances admitting a certified witness. A universal total computable discovery operator is a total computable map \(\mathsf{Discover}\) which, on every \(I\in\mathrm{DISC}_T\), outputs a certified witness for \(I\); its behavior on \(I\notin\mathrm{DISC}_T\) is unrestricted. Determine the computability-theoretic status of \(\mathrm{DISC}_T\) and of the existence of such a universal total computable \(\mathsf{Discover}\). In particular: (1) prove a sharp impossibility theorem showing that certified algorithmic improvements can exist uniformly on a nontrivial class while no total computable discovery operator can produce them; (2) determine whether \(\mathrm{DISC}_T\) is computable and, where possible, establish its exact arithmetical-hierarchy degree; (3) characterize how the answer changes when \(T\) is strengthened or restricted and when \(\Delta\) is restricted to standard speedup regimes; (4) identify nontrivial classes of decision problems, proof systems, or bounded synthesis spaces for which discovery becomes computable or admits explicit upper bounds; and (5) relate the obstruction precisely to Blum speedup, semantic undecidability, proof-search complexity, and modern algorithm-synthesis systems. The core theorem uses only formal correctness, certified asymptotic improvement, and effective discovery. A separate optional extension may impose a decidable syntactic transformation closure \(\mathcal E\) and require \(A\) to lie outside the \(\mathcal E\)-closure of \(K\), but structural novelty is not part of the primary theorem.

Let \(RM(7,14)\) be the binary Reed–Muller code obtained by evaluating all Boolean polynomials \(f:\mathbb F_2^{14}\to\mathbb F_2\) of algebraic degree at most \(7\) on all points of \(\mathbb F_2^{14}\). Thus every codeword has length \(2^{14}=16384\). Define the weight spectrum \[ W_{7,14}=\{\operatorname{wt}(f): f\in RM(7,14)\}, \] where \(\operatorname{wt}(f)=|\{x\in\mathbb F_2^{14}:f(x)=1\}|\). Determine \(W_{7,14}\) exactly. In particular, settle every weight left unresolved by the 2026 work of Leuenberger and Albrizzio. Their remaining construction gap is reduced to the eight representative weights \(322,326,330,334,354,4378,4380,4382\): for each unresolved case, prove occurrence by constructing a codeword of that Hamming weight or prove non-occurrence in \(RM(7,14)\), and combine these decisions with the already established parts of the spectrum to give the complete set \(W_{7,14}\).

Given a graph $F$, a family $\mathcal F$ of graphs on $[n]$ is \emph{$F$-intersecting} if $G\cap H$ contains a copy of $F$ for every $G,H\in\mathcal F$. there exists an absolute constant $\varepsilon>0$ such that every $P_4$-intersecting family $\mathcal F$ satisfies $|\mathcal F|\le\left(\frac12-\varepsilon\right)2^{\binom n2}$, which resolves a conjecture of Alon.

For \(v=(m,n)\in\mathbb Z^2\setminus\{0\}\), write \(r_v=(m^2+n^2)^{1/2}\) and \(\theta_v=\arg(m+in)\). Define \[ C_0(s)=\sum_{v\ne0}r_v^{-2s}=4\zeta(s)\beta(s) \] for \(\Re s>1\), with meromorphic continuation, and for each integer \(k\ge1\), define the angular lattice sum \[ C_k(s)=\sum_{v\ne0}\frac{\cos(4k\theta_v)}{r_v^{2s}}, \] again continued meromorphically. Let \(f_k(\theta)=1-\cos(4k\theta)\), and for sufficiently small real \(\varepsilon\) define an axis-calibrated Pythagorean perturbation by \[ \ell_{\varepsilon,k}(r,\theta)^2=r^2\bigl(1+\varepsilon f_k(\theta)\bigr). \] For fixed \(k\), sufficiently small \(|\varepsilon|\) makes this positive and strictly convex. If \[ Z_{\varepsilon,k}(s)=\sum_{v\ne0}\ell_{\varepsilon,k}(v)^{-2s} \] and \[ P_{\varepsilon,k}(s)=\sum_{\gcd(m,n)=1}\ell_{\varepsilon,k}(m,n)^{-2s}, \] then for \(\Re s>1\), homogeneity gives \(Z_{\varepsilon,k}(s)=\zeta(2s)P_{\varepsilon,k}(s)\), and the first variation satisfies \[ \left.\partial_\varepsilon P_{\varepsilon,k}(s)\right|_{\varepsilon=0} =\frac{-s\,[C_0(s)-C_k(s)]}{\zeta(2s)}. \] Main problem: prove or refute \[ \boxed{\forall\rho\;[\zeta(\rho)=0,\ 0<\Re\rho<1]\Longrightarrow\exists k\ge1:\ C_k(\rho/2)\ne C_0(\rho/2).} \] Equivalently, prove or refute that no nontrivial Riemann zero is simultaneously invisible to every Pythagorean probe \(f_k\). A stronger subproblem asks whether the single mode \(k=1\) is universal: \[ C_1(\rho/2)\ne C_0(\rho/2)\quad\text{for every nontrivial zero }\rho. \] For the intended contradiction strategy toward RH, the logically sufficient detection statement is weaker: it is enough to prove the displayed existential non-cancellation only for hypothetical zeros with \(\Re\rho>1/2\). If a zero already on the critical line were invisible to every \(k\), the stronger all-zero detection statement would fail, but that fact alone would not rule out the RH strategy. To obtain an RH contradiction from detection one would still need a separate, independently proved primitive-response stability theorem excluding the corresponding singularities in \(\Re s>1/4\); no such stability estimate is assumed here.

Let $p_n$ denote the $n$-th odd prime, so $p_1=3,p_2=5,p_3=7,\ldots$. Define $Q(n)=(p_{n+1}-p_n)/2$ for $n\ge1$. For positive integers $x,y,z$, call $(x,y,z)$ a good triple if at least two of $x,y,z$ are equal, or if they are pairwise distinct and $\{x,y,z\}=\{k,k+1,k+2\}$ for some integer $k\ge1$. For $1\le a\le b$, define $Q(a,b)=(Q(a),Q(a+1),\ldots,Q(b))$, with length $b-a+1$. A segment of length at least $3$ is a Vô sequence if every overlapping consecutive triple $(Q(i),Q(i+1),Q(i+2))$, $a\le i\le b-2$, is a good triple. The Vô Sequence Conjecture (VSC) asserts that every Vô sequence has length at most $24$, and that a Vô sequence has length $24$ if and only if $(a,b)=(1,24)$. Equivalently, $Q(1,24)$ is the unique Vô sequence of maximum possible length. Here $Q(1,24)=(1,1,2,1,2,1,2,3,1,3,2,1,2,3,3,1,3,2,1,3,2,3,4,2)$.

Fix a positive exponent parameter smaller than one hundredth. Consider the fixed smoothly forced whole-space incompressible Navier–Stokes singular solution described in the cited September 2026 source, normalized so that its singular time is one. Measure time by the remaining time before the singularity. The singular core has radial size proportional to the square root of the remaining time, axial size proportional to the remaining time raised to one half minus the exponent parameter, dominant azimuthal and axial speeds proportional to the remaining time raised to minus one half minus that parameter, and radial speed no larger in order than the inverse square root of the remaining time. For each sufficiently small positive low-Mach parameter, consider the standard barotropic compressible Navier–Stokes equations in three spatial dimensions with fixed positive shear viscosity, bulk viscosity satisfying the usual nonnegative dissipation condition, and a smooth pressure law whose derivative equals one at unit density and stays positive near unit density. Scale the pressure force by the inverse square of the low-Mach parameter. Use exactly the same smooth compactly supported external force as in the fixed incompressible singular solution, without retuning that force as the low-Mach parameter varies. Start from zero velocity and from a density equal to one plus a perturbation of quadratic size in the low-Mach parameter, with the perturbations uniformly bounded in one fixed Sobolev space of order at least six and with the density uniformly positive. The critical exponent is defined to be two divided by one plus twice the exponent parameter. For every smaller exponent, determine whether the initial density perturbation can be chosen so that the compressible solution remains smooth until the time whose distance from the singular time is the low-Mach parameter raised to that exponent, and so that the compressible velocity shadows the incompressible singular profile throughout the shrinking anisotropic core. Shadowing means that the radial velocity error, multiplied by the square root of the remaining time, tends uniformly to zero, while the azimuthal and axial velocity errors, each multiplied by the remaining time raised to one half plus the exponent parameter, also tend uniformly to zero on every fixed multiple of the shrinking radial and axial core scales. If precritical shadowing holds for every exponent below the critical exponent, determine whether the critical regime, where the remaining time equals the low-Mach parameter raised to that critical exponent, has a nontrivial compressible transition limit after rescaling space by the radial and axial core lengths. In that limit, require the low-Mach parameter times the velocity magnitude to remain of order one. Determine whether the limiting compressible dynamics remains smooth or develops a different singularity. If shadowing fails before the critical regime, identify the earliest failure exponent, or the sharpest rigorous bound on it, and provide a mathematical mechanism for the failure inside this fixed barotropic model. Mere deterioration of estimates known only on fixed time intervals does not count as a negative resolution. The parameter range used here is \((0,1/100)\).

Working over ZF set theory, for a fixed field $K$, does the assertion that every vector space over $K$ has a basis imply the full Axiom of Choice? Determine for which fixed fields $K$ this implication holds. A basis is a linearly independent spanning subset, so every vector is a finite linear combination of basis elements.

For subsets $A,B\subseteq\mathbb N$, let $A\le_{SJT}B$ mean that for every computable nondecreasing unbounded function $h:\mathbb N\to\mathbb N\setminus\{0\}$ there is a uniformly $B$-computably enumerable sequence of finite sets $(T_n)$ with $|T_n|\le h(n)$ such that $J^A(n)\downarrow$ implies $J^A(n)\in T_n$, where $J^A$ is a fixed universal partial $A$-computable jump function. Let $\mathrm{MLR}^X$ denote the Martin-Lof random reals relative to $X$. Must $A\le_{SJT}B$ imply $\mathrm{MLR}^B\subseteq\mathrm{MLR}^A$?

Does there exist a partial computable map $f:\subseteq 2^\omega\to 2^\omega$ that is injective on its domain, takes each Martin-Lof random real in its domain to a Martin-Lof random real, and satisfies $\mu(\operatorname{dom}f)>0$ and $\mu(\{x\in\operatorname{dom}f:f(x)<_T x\})>0$? Here $\mu$ is the fair-coin product measure, and $f(x)<_T x$ means that $f(x)$ is Turing reducible to $x$ while $x$ is not Turing reducible to $f(x)$.

Let $M$ be a countable transitive model of ZFC. Let $C_M$ be the set of reals in $2^\omega$ that are Cohen-generic over $M$, with its standard Borel structure. For $x,y\in C_M$, put $x\,E_M\,y$ if and only if $M[x]=M[y]$. Is $E_M$ hyperfinite for every such $M$? Explicitly, must there be Borel equivalence relations $F_0\subseteq F_1\subseteq\cdots$ on $C_M$, each having only finite equivalence classes, such that $E_M=\bigcup_{n\in\mathbb N}F_n$?

Give $S_\infty$, the group of permutations of $\mathbb N$, the pointwise convergence topology. Call a subgroup $G\le S_\infty$ finitely periodic if for every nonidentity $g\in G$ the cyclic group $\langle g\rangle$ has only finitely many finite orbits on $\mathbb N$. Call $G$ maximal finitely periodic if no proper supergroup of $G$ in $S_\infty$ is finitely periodic. Can such a maximal group be countable, a countable union of compact sets, or Borel? Determine existence in each of these three classes.

Does ZFC prove that there exists a set $E\subseteq\mathbb R^2$ of Hausdorff dimension $1$ such that for every line $L$ through the origin the orthogonal projection of $E$ onto $L$ has Hausdorff dimension $0$? No Borel or analytic regularity is required of $E$.

Let $E$ be equality on $\mathbb R$ and let $F$ be the equivalence relation on $\mathbb R^\mathbb N$ defined by equality of the countable sets enumerated: $x F y$ if and only if $\{x_n:n\in\mathbb N\}=\{y_n:n\in\mathbb N\}$. For equivalence relations on standard Borel spaces, $R\le_B S$ means that some Borel map $f$ satisfies $u R v$ if and only if $f(u) S f(v)$; write $R<_B S$ if $R\le_B S$ and $S\not\le_B R$. Does there exist a complete first-order theory $T$ in a countable language such that $E<_B\cong_T<_B F$, where $\cong_T$ is isomorphism on the standard Borel space of models of $T$ with universe $\mathbb N$?

Determine whether a balanced constant schedule is $s$-composable at every horizon.

$\mathbb{G}^{n+m}$ satisfies $\operatorname{MCP}(K,N)$ if and only if $N\geq n+4m$ and $K\leq 0$.

Let \(p_j\) be the \(j\)-th prime and let \(d_j=p_{j+1}-p_j\). Is it true that every positive integer occurs infinitely often among the values \(\operatorname{lcm}(d_j,d_{j+1})/\gcd(d_j,d_{j+1})\), for \(j\geq1\)?

Let \(p_j\) be the \(j\)-th prime. Is it true that for every integer \(r\geq1\), there is an integer \(j\geq2\) such that \(p_{j+1}-p_j=r(p_j-p_{j-1})\)?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. For a prime $p\equiv1\pmod4$, let $G_p$ be the graph on $\mathbb F_p$ with $u$ adjacent to $v$ exactly when $u-v$ is a nonzero quadratic residue. Let $G_{p,2}$ be the subgraph induced by the common neighbors of $0$ and $1$, and let $\overline{G_{p,2}}$ be its simple graph complement. Is $\vartheta(\overline{G_{p,2}})\le(2/3)\sqrt p$ for every sufficiently large prime $p\equiv1\pmod4$?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. For a prime $p\equiv1\pmod4$, let $G_p$ be the graph on $\mathbb F_p$ with $u$ adjacent to $v$ exactly when $u-v$ is a nonzero quadratic residue. Let $G_{p,1}$ be the subgraph induced by the neighbors of $0$, and let $\overline{G_{p,1}}$ be its simple graph complement on that same vertex set. Is $\vartheta(\overline{G_{p,1}})\sim\sqrt{p/2}$ as $p\to\infty$ through primes congruent to $1$ modulo $4$?

For an integer $M>1$, let $A\in\mathbb R^{(2M-1)\times M}$ be a matrix such that every $M$ rows span $\mathbb R^M$. For $S\subseteq\{1,\ldots,2M-1\}$ let $A_S$ be the submatrix with those rows, and define $\omega(A)=\min_{\operatorname{rank}(A_{S^c})<M}\sigma_M(A_S)$, where $\sigma_M$ is the smallest of its $M$ singular values. Do there exist universal constants $C>0$ and $0<\beta<1$ such that $\omega(A)\le C\beta^M\max_k\|A_k\|_2$ for every such $M,A$?

For each integer $M\ge2$, put $N=4M-5$ and let $A\in\mathbb C^{N\times M}$ have independent standard complex Gaussian entries. Let $p_M$ be the probability that $x\mapsto |Ax|^2$ is injective modulo global complex phase: $|Ax|^2=|Ay|^2$ implies $y=\lambda x$ for some $|\lambda|=1$. Does $\lim_{M\to\infty}p_M=0$?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. Let $G$ be a random circulant graph on $\mathbb Z/n\mathbb Z$. Independently include each unordered difference class $\{k,-k\}$ with $k\ne0$ with probability $1/2$, and join $u,v$ when $u-v$ lies in an included class. Is $\mathbb E\vartheta(G)=(1+o(1))\sqrt n$ as $n\to\infty$?

For a finite simple graph $G$ on $n$ vertices define $\vartheta(G)=\max\{\sum_{i,j}X_{ij}:X\in\mathbb R^{n\times n}, X\succeq0,\operatorname{Tr}(X)=1,\ X_{ij}=0\text{ for every edge }\{i,j\}\}$. Let $G\sim G(n,1/2)$, meaning that each unordered pair of distinct vertices is independently an edge with probability $1/2$. Is $\mathbb E\vartheta(G)=(1+o(1))\sqrt n$ as $n\to\infty$?

Let $F$ be a field whose underlying set is a standard Borel space and whose addition and multiplication maps $F\times F\to F$ are Borel measurable. Must there exist an algebraically closed field $K$ whose underlying set is standard Borel and whose field operations are Borel, together with an injective Borel field homomorphism $F\to K$?

Does there exist a Borel subset $X\subseteq\mathbb R^2$ such that every affine line $L\subseteq\mathbb R^2$ meets $X$ in exactly two points, that is, $|X\cap L|=2$ for every affine line $L$?

Call $A\subseteq\mathbb R^2$ a Kakeya set here if for every real slope $a$ it contains an entire affine line $\{(x,ax+b):x\in\mathbb R\}$ for some $b\in\mathbb R$. Call such a set inclusion-minimal if no proper subset has this property. Does every Borel Kakeya set in this entire-line sense contain an inclusion-minimal Kakeya subset? The minimal subset is not required to be Borel.

Is there an effectively closed ($\Pi^0_1$) class $C\subseteq 2^{\mathbb N}$ such that its computable members form an infinite maximal almost disjoint family among the computable subsets of $\mathbb N$? Each member of that family must be infinite, distinct members have finite intersection, and every infinite computable subset of $\mathbb N$ must have infinite intersection with some computable member of $C$.

Let $M$ be a countably infinite $\omega$-categorical structure in a finite first-order language, and let $G=\operatorname{Aut}(M)$ with its pointwise convergence topology. Let $\operatorname{Aut}(G)$ consist of continuous group automorphisms and let $\operatorname{Inn}(G)$ consist of conjugations by elements of $G$. Must $\operatorname{Out}(G)=\operatorname{Aut}(G)/\operatorname{Inn}(G)$ be finite?

Let \(k>1\), let \(G\) be a group, and let \(a_1H_1,\ldots,a_kH_k\) be pairwise disjoint left cosets of subgroups of finite index in \(G\). Must there be indices \(1\leq i<j\leq k\) such that \(\gcd([G:H_i],[G:H_j])\geq k\)?