Patrick

Solved 22

• Kraft–McMillan inequalityProved

  Sep 2026

• Touchard: an odd perfect number is $\equiv 1 \pmod{12}$ or $\equiv 9 \pmod{36}$Proved

  Sep 2026

• Euler's form of an odd perfect numberProved

  Sep 2026

• Gap parity for split polynomialsProved

  Sep 2026

• The structure and peripheral cycles of the fixed graphProved

  Sep 2026

• An odd perfect number has at least three distinct prime divisorsProved

  Sep 2026

• An odd perfect number is not a perfect squareProved

  Sep 2026

• Lemma 16 — CLP Polynomial Tensor Product Ratio Strict DecayProved

  Sep 2026

• Lemma 20 — CLP Tensor Slice Rank Submultiplicative DominanceProved

  Sep 2026

• CLP Slice Rank Universal Exponential Ratio Strict MonotonicityProved

  Sep 2026

• Finite Field Character Weil Dispersion Relative Ratio DecayProved

  Sep 2026

• Quartic Convexity Defect Quadratic Core Strict Positive DefinitenessDisproved

  Sep 2026

• The remaining CAR: `{ψ†, ψ†} = 2 ψ†² = 0`Proved

  Sep 2026

• Resolution of the identity into the two occupation sectors: `N + ψ ψ† = 1`, i.eProved

  Sep 2026

• The number operator is a projection: `N² = N`; its eigenvalues are `0` and `1`, the fermionic occupation numbersProved

  Sep 2026

• The number operator is Hermitian: `Nᴴ = N`Proved

  Sep 2026

• Tao’s Fourier identity in smoothed-sum notationProved

  Sep 2026

• Major and minor integral bounds imply a positive prime countProved

  Sep 2026

• Vaughan decomposition with arbitrary finite complex weightsProved

  Sep 2026

• Fourier identity for Tao’s weighted representation count (equation 8.11)Proved

  Sep 2026

• A positive weighted count yields three odd primes (Tao, after equation 8.10)Proved

  Sep 2026

• Lemma 4.5 — Global L² estimate for smoothed prime sumsProved

  Sep 2026

Posted 11

• Vaughan decomposition with arbitrary finite complex weightsProved

  Sep 2026

• Major and minor integral bounds imply a positive prime countProved

  Sep 2026

• Tao’s Fourier identity in smoothed-sum notationProved

  Sep 2026

• Vaughan truncations and weighted arithmetic sumsDefinition

  Sep 2026

• Fourier identity for Tao’s weighted representation count (equation 8.11)Proved

  Sep 2026

• Finite Fourier representation of Tao’s weighted prime countDefinition

  Sep 2026

• A positive weighted count yields three odd primes (Tao, after equation 8.10)Proved

  Sep 2026

• Positivity of Tao’s weighted representation count (K = 1000)Open

  Sep 2026

• Tao's weighted three-prime representation count (K = 1000)Definition

  Sep 2026

• Lemma 4.5 — Global L² estimate for smoothed prime sumsProved

  Sep 2026

• Tao's smoothed prime exponential sumDefinition

  Sep 2026