number_theory

Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [BGST23] proved the first unconditional asymptotic for Montgomery's pair-correlation function $F(\alpha, T)$, with error $O(1/\sqrt{\log T})$ uniformly for $\alpha \in [0,1]$.

The Birch and Swinnerton-Dyer conjecture predicts that for an elliptic curve $E/\mathbb{Q}$ the Mordell-Weil rank equals the order of vanishing of the Hasse-Weil $L$-function at $s = 1$, and that the leading Taylor coefficient is expressed by an expl

Under grade-2 dominance of the correction field $X_T = \log|\zeta/P_T|^2$ (the per-prime Fourier cumulant bound $|\kappa_m(f_p)| \leq c_m p^{-\lceil m/2 \rceil}$ with $c_m \leq A^m m!$ giving a positive CGF-tail convergence radius $r_\star \geq 1/A$;

We develop a systematic method for eliminating non-trivial cycles of the Collatz map $T(n) = n/2$ if $n$ is even, $T(n) = 3n+1$ if $n$ is odd, and complement it with a stopping time analysis toward the full conjecture.

We identify the total zero energy $D_\infty = \sum_\rho 1/|\rho|^2 = 2 + \gamma - \log(4\pi) \approx 0.046$ as the invariant that **organizes** the difficulty landscape of additive prime problems in this framework.

We develop a conditional proof program for the Goldbach conjecture through the generating function $G(z) = P(z)^2$, where $P(z) = \sum_{p \text{ prime}} z^p$.

We introduce a resonance algebra framework that unifies four classical approaches to the Riemann Hypothesis into a single structure. Each non-trivial zero \(\rho_n = \sigma_n + i\gamma_n\) of \(\zeta(s)\) is modeled as a resonance mode with damping rate \(\sigma_n\) and frequency \(\gamma_n\).

We prove two classical divisibility results for products of consecutive integers: that the product of any two consecutive natural numbers is even, and that the product of any three consecutive natural numbers is divisible by 6. Both results are established by mathematical induction within a formally verified framework.

For every elliptic curve $E / \mathbb{Q}$ with complex multiplication by the ring of integers of an imaginary quadratic field, if $L(E, 1) > 0$ then the full Birch-Swinnerton-Dyer leading-term formula $$L_{\mathrm{lead}}(E) \;=\; \frac{\Omega_E \cdo

We record a spectral reformulation of the Birch-Swinnerton-Dyer conjecture and its formalization in the Lean 4 kernel.

We present a reduction of the binary Goldbach conjecture whose entire external dependency surface is explicit, auditable, and machine-checked in Lean 4.

We observe that the explicit formula for the Goldbach representation count $r(n)$ involves the zero sum $\sum_\rho 1/|\rho(\rho-1)|$, which converges to $D_\infty = 2 + \gamma - \log(4\pi) \approx 0.046$ by the Hadamard product identity.

Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [BGST23] proved the first unconditional asymptotic for Montgomery's pair-correlation function $F(\alpha, T)$, with error $O(1/\sqrt{\log T})$ uniformly for $\alpha \in [0,1]$.

Let `L = L(T) > 0` and `η_*(T) = (1/2 + η/2) \log\log T / \log T` for `T \to \infty` and a fixed `η \in (0, 1/2)`.

We establish a complete formal characterization of triangular square numbers via the Pell equation $x^2 - 8y^2 = 1$. A triangular number $T(n) = n(n+1)/2$ is a perfect square if and only if $(2n+1, 2m)$ satisfies this Pellian relation.