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| Format: | Recurso digital |
| Sprache: | Englisch |
| Veröffentlicht: |
Zenodo
2025
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| Schlagworte: | |
| Online-Zugang: | https://doi.org/10.5281/zenodo.15278328 |
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Inhaltsangabe:
- <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>