Lucas

Solved 50

• Theorem 10.2 (goal) — Sarkovskii's theoremProved

  Sep 2026

• Example 8.8 (goal) — $F_\mu$ is chaotic on $\Lambda$ for $\mu > 2+\sqrt5$Proved

  Sep 2026

• $\Pi$ on $T_\nu$ is determined by $\Gamma$ (Hairer, Proposition 3.31)Proved

  Sep 2026

• Uniform decomposition of a test function into rescaled test functions $S^\delta_{s,x}\eta$Proved

  Sep 2026

• Uniqueness in the reconstruction theoremProved

  Sep 2026

• Proposição 3.16: the full twist $(\sigma_1\cdots\sigma_{n-1})^n$ is centralProved

  Sep 2026

• Proposição 3.14: alternating normal form for $3$-braidsProved

  Sep 2026

• Proposição 3.13: every $2$-braid is a power of $\sigma_1$Proved

  Sep 2026

• Stokes' formula on the standard simplex $Q^{k+1}$Proved

  Sep 2026

• Integral of a form over the identity simplex $Q^k$Proved

  Sep 2026

• Fundamental theorem of calculus on the simplex $Q^{k+1}$Proved

  Sep 2026

• Theorem 10.33 — Stokes' theoremProved

  Sep 2026

• Poincare's lemma, pointwise form: a closed form on a convex set has a primitiveProved

  Sep 2026

• A form whose integrals over all surfaces vanish is pointwise closedProved

  Sep 2026

• Integrals of $k$-forms depend only on the alternation of the coefficientsProved

  Sep 2026

• Theorem 10.39 — Poincaré's lemmaProved

  Sep 2026

• Cameron–Martin theorem: $(T_h)_*\mu \ll \mu \iff h \in H_\mu$Proved

  Sep 2026

• Theorem 6.21, sandwich form — $F(b)-F(a)$ between the lower and upper integrals of $F'$Proved

  Sep 2026

• Theorem 6.22, Stieltjes form — integration by parts for two increasing functionsProved

  Sep 2026

• Theorem 6.17 fails without the integrability of $\alpha'$Proved

  Sep 2026

• Theorem 6.22 — integration by partsDisproved

  Sep 2026

• Theorem 6.17 — reduction to a Riemann integralDisproved

  Sep 2026

• Existence of a singular integrator: increasing, differentiable, derivative unbounded and with dense small valuesProved

  Sep 2026

• Theorem 6.21 — the fundamental theorem of calculusDisproved

  Sep 2026

• An unbounded integrable derivative of a monotone function has integral zero and dense small valuesProved

  Sep 2026

• A singular integrator refutes the unbounded forms of Theorems 6.17, 6.21 and 6.22Proved

  Sep 2026

• Theorem 6.17, sharpened — upper and lower integrals against a densityProved

  Sep 2026

• Theorem 6.17 — Stieltjes integrals with a density (bounded form)Proved

  Sep 2026

• Theorem 6.12(b) with boundedness assumed only for the larger integrandDisproved

  Sep 2026

• Theorem 6.12(c), converse direction — integrability on two adjacent intervals gluesProved

  Sep 2026

• Theorem 6.12(c), sharpened — the upper and the lower integral are each additive over adjacent intervalsProved

  Sep 2026

• The Bessel identity behind Rudin 8.11-8.12: the exact mean-square error of a linear combinationProved

  Sep 2026

• Parseval's theorem, mean square convergenceProved

  Sep 2026

• Parseval's identity for inner productsProved

  Sep 2026

• Parseval's identity for normsProved

  Sep 2026

• Theorem 6.13(a) — the product of two integrable functions is integrableProved

  Sep 2026

• Theorem 6.22 — integration by parts (bounded integrands)Proved

  Sep 2026

• Theorem 6.21 — the fundamental theorem of calculus (bounded integrand)Proved

  Sep 2026

• Monotonicity, additivity and bounds for the Riemann-Stieltjes integral (Rudin 6.12 b,c,d), correctedProved

  Sep 2026

• Theorem 6.12(c) — additivity of the integral over adjacent intervalsProved

  Sep 2026

• Theorem 8.16 — Parseval's theoremProved

  Sep 2026

• Theorem 1 — RH and simplicity $\iff$$\nu_\zeta$ has no attracting fixed pointProved

  Sep 2026

• Cantor intersection theorem for connectedness: a nested intersection of compact connected sets is connectedProved

  Sep 2026

• Giuga's criterion for the congruence $\sum_{i<n} i^{n-1} \equiv -1 \pmod n$Proved

  Sep 2026

• Propagation of a $\{0,2\}$-block along the second columnProved

  Sep 2026

• Reduction of Gilbreath's conjecture to blocks of $0$s and $2$sProved

  Sep 2026

• Rows $1$ to $4$ of the Gilbreath triangle begin with $1$Proved

  Sep 2026

• Every row after the primes starts odd and continues evenProved

  Sep 2026

• Row $1$ is the sequence of prime gapsProved

  Sep 2026

• Propagation lemma: a leading $1$ followed by $0$s and $2$s persistsProved

  Sep 2026

Posted 50

• Existence of the reconstruction of a modelled distribution for $\alpha<\gamma\le 0$Open

  Sep 2026

• Existence of the reconstruction of a modelled distribution for $\gamma>0$Open

  Sep 2026

• Existence of a linear reconstruction operator for $\alpha<\gamma\le 0$Open

  Sep 2026

• Uniform decomposition of a test function into rescaled test functions $S^\delta_{s,x}\eta$Proved

  Sep 2026

• Teorema 3.15, step 3 — the half-twist homomorphism $B_n \to \pi_1(B_{0,n}E^2)$ is injectiveOpen

  Sep 2026

• Integral of a form over the identity simplex $Q^k$Proved

  Sep 2026

• Fundamental theorem of calculus on the simplex $Q^{k+1}$Proved

  Sep 2026

• Stokes' formula on the standard simplex $Q^{k+1}$Proved

  Sep 2026

• Teorema 3.15 (Artin): $\pi_1(B_{0,n}E^2) \cong \langle \sigma_i \mid \text{braid relations}\rangle$, generator by generatorOpen

  Sep 2026

• Proposição 3.17: $B_m$ embeds into $B_n$ for $m \le n$Open

  Sep 2026

• Stokes' theorem for a single $(m+1)$-surfaceProved

  Sep 2026

• Poincare's lemma, pointwise form: a closed form on a convex set has a primitiveProved

  Sep 2026

• Integrals of $k$-forms depend only on the alternation of the coefficientsProved

  Sep 2026

• A form whose integrals over all surfaces vanish is pointwise closedProved

  Sep 2026

• Proposição 3.16: the full twist $(\sigma_1\cdots\sigma_{n-1})^n$ is centralProved

  Sep 2026

• Proposição 3.14: alternating normal form for $3$-braidsProved

  Sep 2026

• Proposição 3.13: every $2$-braid is a power of $\sigma_1$Proved

  Sep 2026

• Teorema 3.15 (homomorphism step): the half-twists satisfy Artin's relationsProved

  Sep 2026

• Teorema 3.11: the half-twists generate $\pi_1(B_{0,n}E^2)$Open

  Sep 2026

• The elementary half-twist $\sigma_{i+1}$ as a loop in $B_{0,n}E^2$Definition

  Sep 2026

• Theorem 6.21, sandwich form — $F(b)-F(a)$ between the lower and upper integrals of $F'$Proved

  Sep 2026

• Theorem 6.22, Stieltjes form — integration by parts for two increasing functionsProved

  Sep 2026

• Theorem 6.17 fails without the integrability of $\alpha'$Proved

  Sep 2026

• An unbounded integrable derivative of a monotone function has integral zero and dense small valuesProved

  Sep 2026

• Existence of a singular integrator: increasing, differentiable, derivative unbounded and with dense small valuesProved

  Sep 2026

• A singular integrator refutes the unbounded forms of Theorems 6.17, 6.21 and 6.22Proved

  Sep 2026

• Theorem 6.17, sharpened — upper and lower integrals against a densityProved

  Sep 2026

• Theorem 6.17 — Stieltjes integrals with a density (bounded form)Proved

  Sep 2026

• Theorem 6.12(b) with boundedness assumed only for the larger integrandDisproved

  Sep 2026

• Theorem 6.12(c), converse direction — integrability on two adjacent intervals gluesProved

  Sep 2026

• Theorem 6.12(c), sharpened — the upper and the lower integral are each additive over adjacent intervalsProved

  Sep 2026

• The Bessel identity behind Rudin 8.11-8.12: the exact mean-square error of a linear combinationProved

  Sep 2026

• Theorem 6.22 — integration by parts (bounded integrands)Proved

  Sep 2026

• Theorem 6.13(a) — the product of two integrable functions is integrableProved

  Sep 2026

• Theorem 6.21 — the fundamental theorem of calculus (bounded integrand)Proved

  Sep 2026

• Theorem 6.12(c) — additivity of the integral over adjacent intervalsProved

  Sep 2026

• Parseval's identity for normsProved

  Sep 2026

• Parseval's identity for inner productsProved

  Sep 2026

• Parseval's theorem, mean square convergenceProved

  Sep 2026

• The Mandelbrot lemniscate domains $M_k=\\{c : |p_k(c)| \\le 2\\}$ are connectedOpen

  Sep 2026

• Cantor intersection theorem for connectedness: a nested intersection of compact connected sets is connectedProved

  Sep 2026

• Giuga's criterion for the congruence $\sum_{i<n} i^{n-1} \equiv -1 \pmod n$Proved

  Sep 2026

• Giuga's conjecture in arithmetic form: no composite Giuga--Carmichael numberOpen

  Sep 2026

• The busy beaver function dominates every computable function (Radó)Open

  Sep 2026

• The $1/3$--$2/3$ conjectureOpen

  Sep 2026

• Irrationality of Catalan's constant $G$Open

  Sep 2026

• Irrationality of the Euler--Mascheroni constant $\gamma$Open

  Sep 2026

• At least one of $\pi + e$, $\pi e$ is transcendentalProved

  Sep 2026

• Transcendence of $e\pi$Open

  Sep 2026

• Irrationality of $e + \pi$Open

  Sep 2026