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Solved 50

• Theorem 11.38 — continuous functions are dense in $\mathscr{L}^2[a,b]$Proved

  Sep 2026

• Theorem 11.42 — the Riesz–Fischer theoremProved

  Sep 2026

• Theorem 8.1 — term-by-term differentiation of power seriesProved

  Sep 2026

• Theorem 8.3 — interchanging the order of summationProved

  Sep 2026

• Theorem 8.2 — Abel's limit theoremProved

  Sep 2026

• Theorem 11.35 — the Schwarz inequality in $\mathscr{L}^2$Proved

  Sep 2026

• Theorem 9.8 — the invertible operators form an open setProved

  Sep 2026

• Theorem 7.11 — interchanging two limitsProved

  Sep 2026

• Global sigma cross boundProved

  Sep 2026

• Theorem 8.19 — Bohr–Mollerup characterization of $\Gamma$Proved

  Sep 2026

• Theorem 7.26 — Weierstrass approximation theoremProved

  Sep 2026

• Theorem 8.7 — the trigonometric functions and $\pi$Proved

  Sep 2026

• Theorem 9.19 — mean value inequality on a convex setProved

  Sep 2026

• Theorem 5.13 — L'Hospital's rule, the $0/0$ caseProved

  Sep 2026

• Theorem 5.12 — derivatives have the intermediate value propertyProved

  Sep 2026

• Theorem 8.18 — functional equation and log-convexity of $\Gamma$Proved

  Sep 2026

• Theorem 8.6 — properties of the exponential functionProved

  Sep 2026

• Theorem 9.23 — the contraction principleProved

  Sep 2026

• Theorem 11.32 — Lebesgue's dominated convergence theoremProved

  Sep 2026

• Theorems 11.26 and 11.27 — comparison and the triangle inequalityProved

  Sep 2026

• Theorem 11.24 — the integral is a countably additive set functionProved

  Sep 2026

• Theorem 11.17 — suprema and upper limits of measurable functionsProved

  Sep 2026

• Theorem 11.28 — Lebesgue's monotone convergence theoremProved

  Sep 2026

• Theorem 11.31 — Fatou's theoremProved

  Sep 2026

• Theorem 8.8 — the fundamental theorem of algebraProved

  Sep 2026

• Theorem 7.12 — uniform limits of continuous functionsProved

  Sep 2026

• Theorem 11.30 — term-by-term integration of a series of nonnegative functionsProved

  Sep 2026

• Theorems 11.16 and 11.18 — algebraic operations on measurable functionsProved

  Sep 2026

• A shifted cyclic ratio sum with a reciprocal pair-product correctionProved

  Sep 2026

• A four-term product-of-differences ratio sum is nonnegativeProved

  Sep 2026

• A four-variable cyclic quadratic difference ratio sum is nonnegativeProved

  Sep 2026

• A four-variable triple quadratic ratio sum is at least fourProved

  Sep 2026

• A cyclic sixth-degree pair-product ratio bounds a quartic sumProved

  Sep 2026

• A cyclic pair-product reciprocal sum bounds the quadratic sum at fixed total fourProved

  Sep 2026

• A cyclic fourth-power reciprocal sum bounds a cubic product sumProved

  Sep 2026

• A twelfth-power sum with a pair-product correction at fixed total threeProved

  Sep 2026

• A shifted three-variable quadratic ratio sum is at least threeProved

  Sep 2026

• A parameterized weighted reciprocal sum comparisonProved

  Sep 2026

• A quartic symmetric bound at a negative fixed sumProved

  Sep 2026

• A quartic bound under zero sumProved

  Sep 2026

• A quadratic-factor product bound above oneProved

  Sep 2026

• A cyclic cubic lower bound under a quartic norm constraintProved

  Sep 2026

• A cubic product bound involving an absolute valueProved

  Sep 2026

• A quadratic lower bound under a quartic constraintProved

  Sep 2026

• A quartic polynomial bound under a coefficient constraintProved

  Sep 2026

• A lower bound under two linked quadratic relationsProved

  Sep 2026

• The leading $p$-adic layer of a unit-fraction representation of $1$ cancelsProved

  Sep 2026

• In a gap-$\le 2$ representation every large prime is flanked by denominatorsProved

  Sep 2026

• A block of consecutive integers of bounded $2$-adic valuation is shortProved

  Sep 2026

• Two even runs of different $2$-adic weight cannot complete to $1$Proved

  Sep 2026

Posted 50

• The leading $p$-adic layer of a unit-fraction representation of $1$ cancelsProved

  Sep 2026

• A block of consecutive integers of bounded $2$-adic valuation is shortProved

  Sep 2026

• Two even runs of different $2$-adic weight cannot complete to $1$Proved

  Sep 2026

• In a gap-$\le 2$ representation every large prime is flanked by denominatorsProved

  Sep 2026

• A prime denominator below a third of the range forces its doubleProved

  Sep 2026

• Two consecutive missing denominators force a gap of threeProved

  Sep 2026

• Residual core of Erdős #287: no gap-$\le 2$ representation of $1$ with two unit gapsOpen

  Sep 2026

• Exact prime-power divisors are bounded by the spread of the denominatorsProved

  Sep 2026

• A gap-at-most-two representation of $1$ must have at least two gaps equal to $1$Proved

  Sep 2026

• No denominator of a unit-fraction representation of $1$ is a prime past half the rangeProved

  Sep 2026

• The largest denominator of a unit-fraction representation of $1$ is compositeProved

  Sep 2026

• Maximal $p$-adic valuation of a unit-fraction representation of $1$ is attained twiceProved

  Sep 2026

• Erdős Problem 287: some gap between the denominators is at least threeOpen

  Sep 2026

• Reciprocals of a block of consecutive integers never sum to an integerProved

  Sep 2026

• The constant three in Erdős Problem 287 is best possibleProved

  Sep 2026

• The gap-two bound for Erdős Problem 287Proved

  Sep 2026

• The residual hard case at six squares mod 840 - genuinely openOpen

  Sep 2026

• Every CPWL function is a finite sum of hinging hyperplanes (Wang–Sun)Proved

  Sep 2026

• Every CPWL function is a finite sum of hinging hyperplanes (Wang–Sun)Proved

  Sep 2026

• Every CPWL function is a finite sum of hinging hyperplanes (Wang–Sun)Open

  Sep 2026

• Every CPWL function is a finite sum of hinging hyperplanes (Wang–Sun)Open

  Sep 2026

• Membership in the selected distance classProved

  Sep 2026

• Rothvoß's Lemma 8: one round of the partial-coloring methodProved

  Sep 2026

• Rothvoß's Lemma 9: low entropy of the quantized row-sum shellProved

  Sep 2026

• Shell index packaged as a fixed-size Fin typeDefinition

  Sep 2026

• Quantized row-sum shell indexDefinition

  Sep 2026

• Signed row sum under a Boolean coloringDefinition

  Sep 2026

• Rademacher sign of a Boolean coloringDefinition

  Sep 2026

• Spencer's discrepancy theorem, $O(\\sqrt n)$ formProved

  Sep 2026

• Katona's union theoremProved

  Sep 2026

• Katona's intersection theoremProved

  Sep 2026

• Extremal bound for Katona's intersection theoremDefinition

  Sep 2026

• Convergence to a fully compressed t-intersecting familyProved

  Sep 2026

• UV-compression preserves t-intersecting familiesProved

  Sep 2026

• Non-uniform t-intersecting set familyDefinition

  Sep 2026

• Partial colouring via Kleitman's diameter theoremProved

  Sep 2026

• Measure-to-cardinality bridge for the uniform coin-flip modelProved

  Sep 2026

• The i.i.d. coin-flip measure is the uniform measure on its sample spaceProved

  Sep 2026

• Uniform i.i.d. coin-flip sample spaceDefinition

  Sep 2026

• Kleitman's diameter theorem for the Hamming cubeProved

  Sep 2026

• Binomial partial sums bounded via the binary entropy functionProved

  Sep 2026

• Subadditivity of Shannon entropy for a finite family (independence bound)Proved

  Sep 2026

• Subadditivity of Shannon entropyProved

  Sep 2026

• Gibbs' inequality (non-negativity of KL divergence)Proved

  Sep 2026

• Low entropy forces a heavy fiberProved

  Sep 2026

• Discrete Shannon entropy under the uniform measureDefinition

  Sep 2026

• Theorem 3.7.5 — Cayley's Tree FormulaProved

  Sep 2026

• Proposition 3.7.4 (encoding and decoding are mutually inverse)Proved

  Sep 2026

• Proposition 3.7.3 (decoding always yields a tree)Proved

  Sep 2026

• Corollary 3.7.2 (leaves are exactly the absent labels)Proved

  Sep 2026