A Rational Euler Constant in Base 7: Fibonacci-Type Convergence in Rational Harmonic Arithmetic
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2025
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| _version_ | 1866901290895802368 |
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| author | Hassanine, Sam |
| author_facet | Hassanine, Sam |
| contents | <p>This work introduces a Fibonacci‑like sequence Eₙ defined by the recurrence<br>Eₙ = Eₙ₋₁ + (1/7) Eₙ₋₂<br>within a base‑7 framework called Rational Harmonic Arithmetic (RHA). Empirically, the ratio Eₙ₊₁∕Eₙ converges to the algebraic constant r ≈ 1.12678, providing a purely rational model of exponential growth. Each term Eₙ is a rational number whose denominator is 7ⁿ and whose numerator increases in jumps at base‑7 digit‑length thresholds tied to primes. Extending prior RHA rationalizations of π and φ, this construction offers finite, discrete analogues to classical constants. Full documentation includes detailed derivations, worked examples, graphical plots of Eₙ, and a generalization to approximate √2.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15357208 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Rational Euler Constant in Base 7: Fibonacci-Type Convergence in Rational Harmonic Arithmetic Hassanine, Sam Euler constant Rational approximation Base 7 arithmetic Fibonacci sequence Rational Harmonic Arithmetic Number theory Cyclic fractions Mathematical constants Exponential growth models <p>This work introduces a Fibonacci‑like sequence Eₙ defined by the recurrence<br>Eₙ = Eₙ₋₁ + (1/7) Eₙ₋₂<br>within a base‑7 framework called Rational Harmonic Arithmetic (RHA). Empirically, the ratio Eₙ₊₁∕Eₙ converges to the algebraic constant r ≈ 1.12678, providing a purely rational model of exponential growth. Each term Eₙ is a rational number whose denominator is 7ⁿ and whose numerator increases in jumps at base‑7 digit‑length thresholds tied to primes. Extending prior RHA rationalizations of π and φ, this construction offers finite, discrete analogues to classical constants. Full documentation includes detailed derivations, worked examples, graphical plots of Eₙ, and a generalization to approximate √2.</p> |
| title | A Rational Euler Constant in Base 7: Fibonacci-Type Convergence in Rational Harmonic Arithmetic |
| topic | Euler constant Rational approximation Base 7 arithmetic Fibonacci sequence Rational Harmonic Arithmetic Number theory Cyclic fractions Mathematical constants Exponential growth models |
| url | https://doi.org/10.5281/zenodo.15357208 |