Egyptian Fractions Telescoping Paper v0.1.0

Initial release of the CF-Egypt bijection paper.

Main contributions:

• Symbolic tuple representation (u, v, i, j) compressing O(a) fractions into O(log b) tuples
• Explicit CF-Egypt bijection formula (Theorem 5)
• Monotonicity explained via paired CF differences
• Extension to irrationals via prefix stability

Paper: egyptian-fractions-telescoping.pdf (attached)
Implementation: https://github.com/popojan/egypt

Giza Convergents Discovery

December 8, 2025 — A Special Day

Key Discoveries

All three Giza pyramids use consecutive convergents of √φ/2:

Pyramid	Ratio	Convergent	Error
Chefren	2/3	4th	4.8%
Menkaure	5/8	5th	1.7%
Cheops	7/11	6th	0.056%

Queen/King Chamber speculation:

• Queen's shaft 9/11 = arithmetic mean of 7/11 and 1/1
• Queen → balance (Ma'at), King → duality (both extremes)
• Culturally plausible, mathematically exact

Game of Life pyramid:

• Birthday gift animation (January 22, 2022)
• Peak at generation 242 = 11² + 11²
• Egyptian fraction: 7/11 = 1/2 + 1/8 + 1/88

Links

• HSM Stack Exchange discussion
• Full documentation

🤖 Generated with Claude Code

v0.1.0: Chebyshev-Eta Discovery

First Release: Computational Mathematics Explorations

Key Results

Chebyshev Integral Theorem Generalizations:

• Discrete: Σ B(n,k) = n
• Continuous: ∫₀ⁿ B(n,k) dk = n
• Hyperbolic: Works for complex offsets k+ib

Dirichlet Eta Connection (NEW):

(1/2πi) ∮ n^{s-1} · B(n,k) dn = -η(s) / (4π)

• B-function pole structure encodes ALL Dirichlet eta values
• s=1 → ln(2), s=2 → π²/12, s=3 → Apéry's constant

Status

🔬 Numerically verified results, not peer-reviewed proofs.

Modules

• Orbit paclet for Wolfram Language
• Session documentation in docs/sessions/

Citation

Use the DOI from Zenodo (will be assigned after this release).

The 1/π Invariant in Chebyshev Polynomial Geometry

Chebyshev Integral Identity Paper v0.1.0

This paper establishes a surprising geometric invariant:

$$\int_{-1}^{1} |T_{k+1}(x) - x \cdot T_k(x)| , dx = 1$$

for all $k \geq 2$, where $T_k$ denotes the Chebyshev polynomial of the first kind.

Key Results

1. Universal Invariant: The integral equals exactly 1 for all k ≥ 2
2. Lobe Structure: The integrand decomposes into k lobes with alternating signs
3. Individual Lobe Areas: Each lobe has area $\frac{1}{\pi}\left(\sin\frac{\pi}{k} + \sin\frac{2\pi}{k}\right)$
4. Primality Connection: For prime k, all internal lobes are "primitive" (bounded by coprime zeros)

Farey Connection

The partial sums connect Chebyshev geometry to Farey sequences and classical number theory.

Contents

• 3-page paper with proofs
• Figures for k = 2, 3, 17, 31 showing lobe structure

🤖 Generated with Claude Code

Primorial Formula Paper v0.1.0

Primorials as Denominators of Alternating Factorial Sums

Proves that the reduced denominator of the alternating sum equals the primorial.

PDF for arXiv submission (pending endorsement).