On the Finiteness of Base-10 Repunit Primes via Cyclotomic Factorization and Modular Divisibility
Fuente:
Zenodo
Saved in:
| Main Author: | |
|---|---|
| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866902021508956160 |
|---|---|
| author | Fathi, Kevin |
| author_facet | Fathi, Kevin |
| contents | <p>We prove that the number of prime values in the base-10 repunit sequence<br>Rn = (10n − 1)/9 is finite. Our approach combines a modular covering system, derived from small primes<br>whose multiplicative orders divide many values of n, with a classical cyclotomic fac-<br>torization analysis of 10n − 1.</p> <p><br>First, we construct a covering system of congruences n ≡ 0 (mod di), where ordpi (10) =<br>di for various primes pi, and show that for over 80% of integers n, the repunit Rn is<br>divisible by at least one such pi. This proves compositeness for the majority of repunit<br>indices.</p> <p><br>Second, we apply a classical result of Zsigmondy to show that for sufficiently large<br>n, the number 10n − 1, and thus Rn, has at least two distinct prime divisors. This<br>eliminates the remaining indices and yields a complete proof that base-10 repunit<br>primes occur only finitely many times.</p> <p><br>Our proof is entirely elementary, using only multiplicative orders, cyclotomic poly-<br>nomials, and known results in prime divisor growth. It confirms longstanding conjec-<br>tural observations based on computational data and provides a complete answer for<br>base-10 repunit primality.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15278328 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | On the Finiteness of Base-10 Repunit Primes via Cyclotomic Factorization and Modular Divisibility Fathi, Kevin Repunits Repunit primes Bit-length growth Logarithmic density Prime rarity Number theory <p>We prove that the number of prime values in the base-10 repunit sequence<br>Rn = (10n − 1)/9 is finite. Our approach combines a modular covering system, derived from small primes<br>whose multiplicative orders divide many values of n, with a classical cyclotomic fac-<br>torization analysis of 10n − 1.</p> <p><br>First, we construct a covering system of congruences n ≡ 0 (mod di), where ordpi (10) =<br>di for various primes pi, and show that for over 80% of integers n, the repunit Rn is<br>divisible by at least one such pi. This proves compositeness for the majority of repunit<br>indices.</p> <p><br>Second, we apply a classical result of Zsigmondy to show that for sufficiently large<br>n, the number 10n − 1, and thus Rn, has at least two distinct prime divisors. This<br>eliminates the remaining indices and yields a complete proof that base-10 repunit<br>primes occur only finitely many times.</p> <p><br>Our proof is entirely elementary, using only multiplicative orders, cyclotomic poly-<br>nomials, and known results in prime divisor growth. It confirms longstanding conjec-<br>tural observations based on computational data and provides a complete answer for<br>base-10 repunit primality.</p> |
| title | On the Finiteness of Base-10 Repunit Primes via Cyclotomic Factorization and Modular Divisibility |
| topic | Repunits Repunit primes Bit-length growth Logarithmic density Prime rarity Number theory |
| url | https://doi.org/10.5281/zenodo.15278328 |