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Math-201-A2 · revision r1 · on Math-201

OpenAI claim summary for Math-201

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Infinitely many primitive-root primes, even with the lower bound $x/(\log x)^2$, do not establish a positive proportion among all primes or its predicted Euler-product value. The simultaneous-prime-base paper also retains four explicit assumptions. General rank-one torus formulations over other number fields exceed the integer-base setting.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Artin's Conjecture on Primitive Roots
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Artin's Conjecture on Primitive Roots

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 029; 2026-10-04), the asserted result is that each integer other than $-1$ or a square is primitive modulo at least $c_a x/(\log x)^2$ primes between $x$ and $2x$ for large $x$; In manuscript 2 (family 029; 2026-10-04), the asserted result is that a fixed finite list of positive prime bases simultaneously has full multiplicative order for at least $cx/(\log x)^2$ primes in a dyadic interval, conditional on four specified companion inputs.

Mathematical statement of the claimed result

Manuscript 1.
An integer $a$ is admissible when $a\ne-1$ and it is not an integer square. For each such $a$, the claim supplies $c_a>0$, $x_a\ge2$ such that every real $x\ge x_a$ satisfies \[ |\{p\text{ prime}:x<p<2x,\ p\nmid a,\ \operatorname{ord}_p(a)=p-1\}| \ge c_a x/(\log x)^2. \] Here $\operatorname{ord}_p(a)$ is multiplicative order. The estimate yields infinitely many primitive-root primes for each admissible base, without asserting Artin's predicted positive-density asymptotic.
Manuscript 2.
For the conditional inputs, let $L=\log x$, $W=\exp(L^{0.24})$, $P^-(h)$ be the least prime factor with $P^-(1)=\infty$, and $V(y)=\prod_{p\le y}(1-1/p)$. Fix $K\ge1$ and $0.1<a_1<\cdots<a_K<0.2$, and set \[ \mathcal P_i=\{p\text{ prime}:e^{L^{a_i}}\le p\le e^{2L^{a_i}}\},\quad V_i=\sum_{p\in\mathcal P_i}1/p,\quad\omega_i(h)=|\{p\in\mathcal P_i:p\mid h\}|, \quad\mathcal W(h)=2^{K-\sum_i\omega_i(h)}\prod_i\frac{\omega_i(h)}{V_i}. \] For $w>\gamma>0$, the density in the inputs is the locally finite sum \[ D_{\gamma}(w)=\frac1w+ \sum_{j\ge2}\frac1{j!} \int_{\substack{t_i\ge\gamma\ (1\le i<j)\\\sum_{i<j}t_i\le w-\gamma}} \frac1{w-\sum_{i<j}t_i}\prod_{i<j}\frac{dt_i}{t_i}. \]

This simultaneous-prime-base theorem is conditional on four separate inputs: zero-freeness of every finite-order Hecke $L_F(s,\chi)$ over every cyclotomic $F\supseteq\mathbb Q(\mu_{12})$ in $\operatorname{Re}s>1-10^{-6}$ (with principal poles allowed); the marked Type-II coefficient-cancellation implication and rough-coefficient cancellation; a uniform block-sieve estimate; and the rough-number density estimate. For every fixed $\delta,C,D_*>0$, the Type-II premise uses coefficients $\alpha_m,\beta_n$, bounded by $L^C$, supported on $W$-rough integers in $[H_m,2H_m]$ and $[H_n,2H_n]$, with $\alpha$ further restricted to any interval $I$. For arbitrary fixed $A_0,B,A>0$, this premise imposes a uniform estimate on all Dirichlet characters $\chi$ with modulus $k\le L^{A_0}$, using frequencies in the range $|t|\le2XL^B$: \[ \left|H_m^{-1}\sum_m\alpha_m\chi(m)m^{it}\right|\ll_A L^{-A}. \] Characters are zero on nonunits. Its conclusion, for every fixed $D_*>0$, is \[ |\sum_{m,n}\alpha_m\beta_nF(mn-1)\mathcal W(mn-1)|\ll X(\log x)^{-D_*}, \] with $H_m,H_n\ge x^{\delta}$, $H_mH_n\asymp x$, $|F|\le1$ and $F(ph)=F(h)$ at group primes. The bound holds for sufficiently large fixed $K$, with the threshold and constants allowed to depend on fixed parameters and the $a_i$. The same scalar cancellation is also assumed for \[ \alpha_m=m^{iv}\left(\mathbf1_{P^-(m)>x^{\gamma}} -\frac{D_{\gamma}(\log m/L)}{LV(W)}\mathbf1_{P^-(m)>W}\right)\mathbf1_{m\in I}, \] for $|v|\le L^C$ and fixed $0<\gamma<w_-<w_+<1$, whenever the logarithmic dyad $[\log H_m/L,\log(2H_m)/L]$ lies in $[w_-,w_+]$.

For a finite ground set with nonnegative weights $\lambda_\nu$, designate primes in a set $\mathcal D\subseteq\{p:p\le z\}$ and corresponding bad subsets $\mathcal A_p$. For a designated squarefree product $\ell$, let $\mathcal A_\ell=\bigcap_{p\mid\ell}\mathcal A_p$, using the whole ground set at $\ell=1$. Require $\sum_{\nu\in\mathcal A_\ell}\lambda_\nu=Yg(\ell)+R(\ell)$, $Y\ge0$, with $g(1)=1$ and $g(p)=0$ outside $\mathcal D$. For these weighted bad-event intersections, the block input assumes $g$ squarefree-multiplicative, $0\le g(p)\le1-\eta_0$ and $\sum_{u<p\le u^2}g(p)\le C_0$. Writing $S=\sum_{\nu\notin\bigcup_{p\in\mathcal D}\mathcal A_p}\lambda_\nu$, the input asserts \[ S=Y\prod_{p\le z}(1-g(p))(1+O(e^{-H})) +O(\sum_{\ell\le z^{4H+2}}|R(\ell)|) \] for large even $H$, summing designated squarefree products including one. The implicit constants and the lower threshold on $H$ depend only on $\eta_0,C_0$, and remain uniform as $H$ increases. The density input requires, for all sufficiently small fixed $b>0$, \[ \int_b^{1/2}D_t(1-t)\,dt/t=D_b(1)-1, \qquad D_b(1)\le e^{-\gamma_E}b^{-1}(1+C_de^{-c_d/b}), \] where $D_b$ is defined above, $\gamma_E$ is Euler's constant, and $c_d,C_d$ and the allowed upper bound on $b$ are absolute. The block input also assumes the upper bound $S\ll Y\prod_{p\le z}(1-g(p))+\sum_{\ell\le z^{10}}|R(\ell)|$ for $H=2$, with the same designated squarefree-product convention. Subject to all four inputs, for any distinct positive prime bases $q_1,\ldots,q_k$, $k\ge1$, the claimed constants $c_{\mathbf q}>0,x_{\mathbf q}$ give \[ |\{p\in(x,2x):p\text{ prime},\ p\nmid\prod_iq_i,\ \operatorname{ord}_p(q_i)=p-1\ (1\le i\le k)\}| \ge c_{\mathbf q}x/(\log x)^2\quad(x\ge x_{\mathbf q}). \] The companion input statements are essential premises, rather than proofs supplied by this manuscript.

Outline of the argument

For manuscript 1 (family 029; 2026-10-04), cubic theta coefficients under Kummer twists give a Hecke zero-free comparison. Controlled prime predecessors in fixed progressions are then sieved against complete-splitting obstructions, retaining enough primes on which the base has full multiplicative order.

For manuscript 2 (family 029; 2026-10-04), uniform splitting estimates remove bad order divisors from a marked family in a progression. Rough-number counts and separation of the least prime factor eliminate balanced composites, leaving the simultaneous prime conclusion under those inputs.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 029; 2026-10-04), manuscript 2 (family 029; 2026-10-04).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources